2.1 Convolution formula and orthogonality relations

As mentioned in the introduction, the present work is concerned with integration on the complex Grasmannian $\mathrm {Gr}(M,N)$ for $M < N$ . We interpret this Grassmannian as the space of $N \times N$ idempotent Hermitian matrices of rank M, which admits three equivalent descriptions as per the following definition.

Definition 2.1 Let $I_M$ denote the $M \times M$ identity matrix and $I_{M,N}$ denote the $N \times N$ matrix whose first M diagonal entries are 1 and whose remaining entries are 0. For $M < N$ , define the space

$$ \begin{align*} \mathbf{S}(M,N) &= \{ S \in \mathrm{Mat}_{N \times N}(\mathbb{C}) \mid S^2 = S, S = S^* \text{ and } \mathrm{rank}(S) = M \} \\ &= \{ S = U^*U \mid U \in \mathrm{Mat}_{M \times N}(\mathbb{C}) \text{ and } UU^* = I_M \} \\ &= \{ S = U I_{M,N} U^* \mid U \in \mathbf{U}(N) \}. \end{align*} $$

The unitary group $\mathbf {U}(N)$ acts transitively on $\mathbf {S}(M,N)$ by conjugation, thus endowing it with the structure of a compact homogeneous space. Thus, the Haar measure on $\mathbf {U}(N)$ induces a $\mathbf {U}(N)$ -invariant normalised Haar measure on $\mathbf {S}(M,N)$ , which we denote succinctly by $\mathrm {d} S$ . (See [Reference Diestel and Spalsbury16] for an introduction to Haar measures.) The fact that the stabilizer of $I_{M,N} \in \mathbf {S}(M,N)$ is $\mathbf {U}(M) \times \mathbf {U}(N-M)$ allows us to identify $\mathbf {S}(M,N)$ with the complex Grassmannian $\mathrm {Gr}(M,N) \cong \mathbf {U}(N) \, / \, \mathbf {U}(M) \times \mathbf {U}(N-M)$ .

For $1 \leqslant i, j \leqslant N$ , define the function $S_{ij}: \mathbf {S}(M,N) \to \mathbb {C}$ corresponding to the $(i,j)$ matrix element. Our primary goal is to calculate integrals of the form

$$\begin{align*}\int_{\mathbf{S}(M,N)} S_{i_1 j_1} S_{i_2 j_2} \cdots S_{i_k j_k} \, \mathrm{d} S, \end{align*}$$

where $1 \leqslant i_1, i_2, \ldots , i_k, j_1, j_2, \ldots , j_k \leqslant N$ . We impose the technical assumption that $k \leqslant N$ for future convenience. However, this assumption has little bearing on our work, since we will study these matrix integrals in the regime of large N and fixed ratio $\frac {M}{N}$ . The following elementary integrals will be of particular importance.

Definition 2.2 (Weingarten function)

For each permutation $\sigma \in S_k$ , define the integral

$$\begin{align*}{\mathrm{Wg}^{\mathbf{S}}}(\sigma) = \int_{\mathbf{S}(M,N)} S_{1,\sigma(1)} S_{2,\sigma(2)} \cdots S_{k,\sigma(k)} \, \mathrm{d} S. \end{align*}$$

We refer to the function ${\mathrm {Wg}^{\mathbf {S}}}: S_0 \sqcup S_1 \sqcup S_2 \sqcup \cdots \to \mathbb {C}$ as the Weingarten function for $\mathbf {S}(M,N)$ . Here, we include the symmetric group $S_0$ , whose unique element is denoted $(\,)$ and represents the empty permutation. In the notation for the Weingarten function, we suppress the dependence on M and N to avoid clutter.

The setup described above sits firmly in the realm of Weingarten calculus, which is broadly concerned with the calculation of integrals on compact groups and related objects with respect to the Haar measure [Reference Collins, Matsumoto and Novak14]. Modern accounts of Weingarten calculus often rely on elegant algebraic approaches via Schur–Weyl duality [Reference Collins12, Reference Collins and Śniady15]. A direct use of such an argument is not immediately available for the case of integration over $\mathbf {S}(M,N)$ . We instead follow the approach via orthogonality relations utilized by Collins and Matsumoto [Reference Collins and Matsumoto13], which is in turn inspired by the ideas contained in the seminal paper of Weingarten [Reference Weingarten49].

In the remainder of this section, we develop the Weingarten calculus for integration on $\mathbf {S}(M,N)$ in three parts. First, we prove a convolution formula that reduces general integrals of monomials in the matrix elements to the elementary ones defined in Definition 2.2. Second, we prove so-called orthogonality relations that completely determine all values of the Weingarten function. Third, we solve the linear system provided by these orthogonality relations, which connects naturally to the representation theory of the symmetric groups, particularly to the Jucys–Murphy elements in the symmetric group algebra.

Theorem 2.3 (Convolution formula)

Arbitrary integrals of monomials in the matrix elements of $\mathbf {S}(M,N)$ reduce to elementary integrals via the equation

$$\begin{align*}\int_{\mathbf{S}(M,N)} S_{i_1j_1} S_{i_2j_2} \cdots S_{i_kj_k} \, \mathrm{d} S = \sum_{\sigma \in S_k} \delta_{i_{\sigma(1)},j_1} \delta_{i_{\sigma(2)},j_2} \cdots \delta_{i_{\sigma(k)},j_k} {\mathrm{Wg}^{\mathbf{S}}}(\sigma). \end{align*}$$

Proof Write $S \in \mathbf {S}(M,N)$ in the form $S = UI_{M,N}U^*$ for $U \in \mathbf {U}(N)$ . Then the two sides of the desired equation can be equivalently expressed as integrals over $\mathbf {U}(N)$ .

$$ \begin{align*} \text{LHS} &= \sum_{m_1, \ldots, m_k = 1}^N \bigg(\prod_{i=1}^k \left[I_{M,N}\right]_{m_im_i} \bigg) \int_{\mathbf{U}(N)} U_{i_1m_1} \cdots U_{i_km_k} U_{m_1j_1}^* \cdots U_{m_kj_k}^* \, \mathrm{d} U \\ \text{RHS} &= \sum_{\sigma \in S_k} \bigg( \prod_{a=1}^k \delta_{i_{\sigma(a)},j_a} \bigg) \sum_{m_1, \ldots, m_k = 1}^N \bigg(\prod_{i=1}^k \left[I_{M,N}\right]_{m_im_i} \bigg) \\ & \qquad \qquad \qquad \qquad \qquad \quad~ \int_{\mathbf{U}(N)} U_{1m_1} \cdots U_{km_k} U_{m_1,\sigma(1)}^* \cdots U_{m_k,\sigma(k)}^* \, \mathrm{d} U.\end{align*} $$

Thus, the result would follow from the equation

$$ \begin{align*} \int_{\mathbf{U}(N)} &U_{i_1m_1} \cdots U_{i_km_k} U_{m_1j_1}^* \cdots U_{m_kj_k}^* \, \mathrm{d} U \\ &= \sum_{\sigma \in S_k} \delta_{i_{\sigma(1)},j_1} \cdots \delta_{i_{\sigma(k)},j_k} \int_{\mathbf{U}(N)} U_{1m_1} \cdots U_{km_k} U_{m_1,\sigma(1)}^* \cdots U_{m_k,\sigma(k)}^* \, \mathrm{d} U. \end{align*} $$

However, this is a direct consequence of applying the convolution formula for $\mathbf {U}(N)$ [Reference Collins and Śniady15, Corollary 2.4] to both sides.

The following orthogonality relations for ${\mathrm {Wg}^{\mathbf {S}}}$ are inspired by the orthogonality relations obtained by Collins and Matsumoto in other settings for Weingarten calculus [Reference Collins and Matsumoto13]. The proof crucially relies on the convolution formula of Theorem 2.3. For future reference, observe that we use the notational convention of composing permutations from right to left.

Theorem 2.4 (Orthogonality relations)

For each permutation $\sigma \in S_k$ , the Weingarten function satisfies the relation

$$ \begin{align*} {\mathrm{Wg}^{\mathbf{S}}}(\sigma) ={}& -\frac{1}{N} \sum_{i=1}^{k-1} {\mathrm{Wg}^{\mathbf{S}}}(\sigma \circ (i~k)) + \delta_{\sigma(k),k} \, \frac{M}{N} {\mathrm{Wg}^{\mathbf{S}}}(\sigma^\downarrow) \\ &+ \frac{1}{N} \sum_{i=1}^{k-1} \delta_{\sigma(i),k} {\mathrm{Wg}^{\mathbf{S}}}([\sigma \circ (i~k) ]^\downarrow). \end{align*} $$

Here, for any permutation $\rho \in S_k$ that fixes k, we write $\rho ^\downarrow \in S_{k-1}$ to denote the permutation that agrees with $\rho $ on the set $\{1, 2, \ldots , k-1\}$ .

Proof Let $\sigma \in S_k$ and consider the cases $\sigma (k) = k$ and $\sigma (k) \neq k$ separately.

Case 1. Suppose $\sigma (k) = k$ .

Consider the integral

(*)

On the one hand, we can use $\sum _{i=1}^N S_{ii} = \mathrm {Tr}(S) = \mathrm {Tr}(UI_{M,N}U^*) = \mathrm {Tr}(I_{M,N}) = M$ and the definition of ${\mathrm {Wg}^{\mathbf {S}}}$ to express the integral as $M {\mathrm {Wg}^{\mathbf {S}}}(\sigma ^\downarrow )$ . On the other hand, we can apply the convolution formula directly to each summand. For the ith summand, where $1 \leqslant i < k$ , the convolution formula yields

$$\begin{align*}\int_{ \mathbf{S}(M,N) } S_{1,\sigma(1)} \cdots S_{i,\sigma(i)} \cdots S_{k-1,\sigma(k-1)} S_{i,i} \, \mathrm{d} S = {\mathrm{Wg}^{\mathbf{S}}}(\sigma) + {\mathrm{Wg}^{\mathbf{S}}}(\sigma \circ (i~k)). \end{align*}$$

For the ith summand, where $k \leqslant i \leqslant N$ , the convolution formula yields

$$\begin{align*}\int_{ \mathbf{S}(M,N) } S_{1, \sigma(1)} S_{2, \sigma(2)} \cdots S_{k-1, \sigma(k-1)} S_{i,i} \, \mathrm{d} S = {\mathrm{Wg}^{\mathbf{S}}}(\sigma). \end{align*}$$

Adding these contributions over $i = 1, 2, \ldots , N$ and equating with the expression we previously obtained for the integral ( $\ast $ ) leads to

(2.1) $$ \begin{align} M {\mathrm{Wg}^{\mathbf{S}}}(\sigma^\downarrow) &= N {\mathrm{Wg}^{\mathbf{S}}}(\sigma) + \sum_{i=1}^{k-1} {\mathrm{Wg}^{\mathbf{S}}}(\sigma \circ (i~k)) \notag \\ \Rightarrow \qquad {\mathrm{Wg}^{\mathbf{S}}}(\sigma) &= - \frac{1}{N} \sum_{i=1}^{k-1} {\mathrm{Wg}^{\mathbf{S}}}(\sigma \circ (i~k)) + \frac{M}{N} {\mathrm{Wg}^{\mathbf{S}}}(\sigma^\downarrow). \end{align} $$

Case 2. Suppose $\sigma (k) \neq k$ .

Let $j = \sigma ^{-1}(k)$ and consider the integral

(**)

On the one hand, we have $S^2 = S$ for all $S \in \mathbf {S}(M,N)$ , so it follows that $\sum _{i=1}^{N} S_{j,i} S_{i, \sigma (k)} = S_{j, \sigma (k)}$ . Combining this observation with the convolution formula allows us to express the integral as follows:

$$\begin{align*}\int_{\mathbf{S}(M,N)} S_{1, \sigma(1)} \cdots S_{j-1, \sigma(j-1)} S_{j, \sigma(k)} S_{j+1, \sigma(j+1)} \cdots S_{k-1, \sigma(k-1)} \, \mathrm{d} S = {\mathrm{Wg}^{\mathbf{S}}}(\left[\sigma \circ (j~k)\right]^\downarrow). \end{align*}$$

On the other hand, we can apply the convolution formula directly to each summand, resulting in a calculation analogous to that of Case 1. For the ith summand, where $1 \leqslant i < k$ , the convolution formula yields

$$ \begin{align*} \int_{\mathbf{S}(M,N)} \left( S_{1, \sigma(1)} \cdots S_{j-1, \sigma(j-1)} \right) S_{j,i} &\left( S_{j+1, \sigma(j+1)} \cdots S_{k-1, \sigma(k-1)} \right) S_{i, \sigma(k)} \, \mathrm{d} S \\ &\quad\qquad\qquad = {\mathrm{Wg}^{\mathbf{S}}}(\sigma) + {\mathrm{Wg}^{\mathbf{S}}}(\sigma \circ (i~k)). \end{align*} $$

For the ith summand, where $k \leqslant i \leqslant N$ , the convolution formula yields

$$\begin{align*}\int_{\mathbf{S}(M,N)} \left( S_{1, \sigma(1)} \cdots S_{j-1, \sigma(j-1)} \right) S_{j,i} \left( S_{j+1, \sigma(j+1)} \cdots S_{k-1, \sigma(k-1)} \right) S_{i, \sigma(k)} \, \mathrm{d} S = {\mathrm{Wg}^{\mathbf{S}}}(\sigma). \end{align*}$$

Adding these contributions over $i = 1, 2, \ldots , N$ and equating with the expression we previously obtained for the integral ( $\ast \ast $ ) leads to

(2.2) $$ \begin{align} {\mathrm{Wg}^{\mathbf{S}}}(\left[\sigma \circ (j~k)\right]^\downarrow) &= N {\mathrm{Wg}^{\mathbf{S}}}(\sigma) + \sum_{i=1}^{k-1} {\mathrm{Wg}^{\mathbf{S}}}(\sigma \circ (i~k)) \notag \\ \Rightarrow \qquad {\mathrm{Wg}^{\mathbf{S}}}(\sigma) &= -\frac{1}{N} \sum_{i=1}^{k-1} {\mathrm{Wg}^{\mathbf{S}}}(\sigma \circ (i~k)) + \frac{1}{N} {\mathrm{Wg}^{\mathbf{S}}}(\left[\sigma \circ (j~k)\right]^\downarrow). \end{align} $$

Finally, the desired result is obtained by writing the two expressions for ${\mathrm {Wg}^{\mathbf {S}}}(\sigma )$ obtained in equations (2.1) and (2.2) from the two separate cases in one formula, making use of the Kronecker delta notation.

The orthogonality relations provide a non-degenerate linear system of equations that uniquely determines the Weingarten function. The example below shows that values of the Weingarten function can be computed explicitly and are rational functions of M and N.

Example 2.5 By the orthogonality relations of Theorem 2.4 and the conjugacy invariance of ${\mathrm {Wg}^{\mathbf {S}}}$ , we obtain the following equations.

$$ \begin{align*} {\mathrm{Wg}^{\mathbf{S}}}((1)(2)(3)) &= -\frac{1}{N} \left[ {\mathrm{Wg}^{\mathbf{S}}}((1~3)(2)) + {\mathrm{Wg}^{\mathbf{S}}}((2~3)(1)) \right] + \frac{M}{N} {\mathrm{Wg}^{\mathbf{S}}}((1)(2)) \\ &= -\frac{2}{N} {\mathrm{Wg}^{\mathbf{S}}}((1~2)(3)) + \frac{M}{N} {\mathrm{Wg}^{\mathbf{S}}}((1)(2)) \\ {\mathrm{Wg}^{\mathbf{S}}}((1~2)(3)) &= -\frac{1}{N} \left[ {\mathrm{Wg}^{\mathbf{S}}}((1~3~2)) + {\mathrm{Wg}^{\mathbf{S}}}((1~2~3) \right] + \frac{M}{N} {\mathrm{Wg}^{\mathbf{S}}}((1~2)) \\ &= -\frac{2}{N} {\mathrm{Wg}^{\mathbf{S}}}((1~2~3)) + \frac{M}{N}{\mathrm{Wg}^{\mathbf{S}}}((1~2)) \\ {\mathrm{Wg}^{\mathbf{S}}}((1~2~3)) &= -\frac{1}{N} \left[ {\mathrm{Wg}^{\mathbf{S}}}((2~3)(1)) + {\mathrm{Wg}^{\mathbf{S}}}((1~2)(3) \right] + \frac{1}{N} {\mathrm{Wg}^{\mathbf{S}}}((1~2)) \\ &= -\frac{2}{N} {\mathrm{Wg}^{\mathbf{S}}}((1~2)(3)) + \frac{1}{N} {\mathrm{Wg}^{\mathbf{S}}}((1~2)). \end{align*} $$

Using the values ${\mathrm {Wg}^{\mathbf {S}}}((1)(2)) = \frac {M(MN-1)}{N (N^2-1)}$ and ${\mathrm {Wg}^{\mathbf {S}}}((12)) = \frac {-M(M - N)}{N (N^2-1)}$ , one can solve this linear system of equations to obtain the following unique solution.

$$ \begin{align*} {\mathrm{Wg}^{\mathbf{S}}}((1)(2)(3)) &= \frac{-2(MN-2)}{N(N^2-4)} {\mathrm{Wg}^{\mathbf{S}}}((1~2)) + \frac{M}{N} {\mathrm{Wg}^{\mathbf{S}}}((1)(2)) \\ &= \frac{M(M^2N^2 - 2M^2 - 3MN + 4)}{N (N^2-1) (N^2-4)} \\ {\mathrm{Wg}^{\mathbf{S}}}((1~2)(3)) &= \frac{MN-2}{N^2-4} {\mathrm{Wg}^{\mathbf{S}}}((1~2)) = \frac{-M (M-N) (MN-2)}{N (N^2-1) (N^2-4)} \\ {\mathrm{Wg}^{\mathbf{S}}}((1~2~3))) &=\frac{N-2M}{N^2-4}{\mathrm{Wg}^{\mathbf{S}}}((1~2)) = \frac{M (M-N) (2M-N)}{N (N^2-1) (N^2-4)}. \end{align*} $$

In this way, one can begin with the base case ${\mathrm {Wg}^{\mathbf {S}}}((\,)) = 1$ and inductively obtain ${\mathrm {Wg}^{\mathbf {S}}}(\sigma )$ for $\sigma \in S_k$ in terms of ${\mathrm {Wg}^{\mathbf {S}}}(\sigma ')$ for $\sigma '\in S_{k-1}$ . Further values can be found in Appendix A.

2.2 Large N expansion

A priori, computing the values of the Weingarten function ${\mathrm {Wg}^{\mathbf {S}}}$ via the integral definition is far from straightforward. However, as evidenced by the calculations of Example 2.5, the orthogonality relations of Theorem 2.4 uniquely determine the Weingarten function and imply that its values are rational functions of M and N. We will shortly see that their structure also leads directly to a large N expansion for ${\mathrm {Wg}^{\mathbf {S}}}(\sigma )$ , with coefficients that enumerate factorizations of $\sigma $ into transpositions that satisfy a certain monotonicity condition. This combinatorial structure can be understood in terms of paths in the Weingarten graph, which encode the result of recursively applying the orthogonality relations ad infinitum. The notion of a Weingarten graph was originally introduced by Collins and Matsumoto [Reference Collins and Matsumoto13]. In the setting of integration over unitary groups, the Weingarten graph $\mathcal {G}^{\mathbf {U}}$ has two types of edges, which reflects the fact that the orthogonality relations in that case express the Weingarten function of a permutation in terms of two types of terms. In the setting of integration over $\mathbf {S}(M,N)$ , we have three terms appearing on the right side of the orthogonality relations, motivating the following definition (Figure 1).

Figure 1: The part of the Weingarten graph $\mathcal {G}^{\mathbf {S}}$ induced by vertices belonging to $S_0 \sqcup S_1 \sqcup S_2 \sqcup S_3$ .

Repeated application of the orthogonality relations leads to paths in the Weingarten graph $\mathcal {G}^{\mathbf {S}}$ . In this way, one is motivated to enumerate paths in the Weingarten graph in which edges of types A, B, C receive multiplicative weights $-\frac {1}{N}$ , $\frac {M}{N}$ , $\frac {1}{N}$ , respectively.

Definition 2.8 A path in $\mathcal {G}^{\mathbf {S}}$ is a sequence of permutations

where $(\rho _{i-1}, \rho _{i})$ is a directed edge of $\mathcal {G}^{\mathbf {S}}$ for each $i = 1, 2, \ldots , \ell $ . We call the integer

the length of the path and denote by $\mathcal {P}(\sigma , \sigma ^\prime )$ the set of all paths from $\sigma $ to $\sigma '$ . Define the weight

of a path

by

where

denotes the number of edges of type $K\in \{A,B,C\}$ on the path

.

By construction, the Weingarten graph is a combinatorial encoding of the orthogonality relations and their repeated application. Starting with $\sigma \in S_k$ , iterating the orthogonality relations $\ell $ times produces the formula

By sending the number of iterations $\ell $ to infinity, one obtains the following large N expansion of the Weingarten function.

Proposition 2.9 For any permutation $\sigma $ , we have the large N expansion

At this point, let us consider the Weingarten function in the regime of large N with fixed ratio $q = \frac {M}{N}$ .Footnote ² For

and $\sigma \in S_k$ , we have

, which leads to

To develop a clearer combinatorial description of the coefficients appearing in the expansion above, we consider the correspondence between paths in the Weingarten graph and monotone factorizations. This connection already appears in the Weingarten calculus for unitary groups [Reference Collins and Matsumoto13].

Given a path in the Weingarten graph $\mathcal {G}^{\mathbf {S}}$ from $\sigma $ to $(\,)$ , one can record the sequence of transpositions arising from the type A edges and the type C edges. The composition of these transpositions in reverse order recovers the permutation $\sigma $ . Thus, a path gives rise to a monotone factorization satisfying . Represent this construction via the map

$$\begin{align*}\mathcal{F}: \mathcal{P}(\sigma,(\,)) \to \mathcal{M}(\sigma). \end{align*}$$

In the analogous construction for the unitary case, one obtains a one-to-one correspondence between paths and monotone factorizations, but that is not the case here. Given a monotone factorization

of $\sigma $ , there exists a unique path in

containing only edges of types A and B. The number of pairs of consecutive A–B edges in the path is equal to the hive number

. Any such pair can be replaced by a type C edge to obtain an element of

and every element of

can be obtained in this way. Thus, we have

. Moreover, keeping track of the effect on edge weights, we have the equality

These observations allow us to express the coefficients of the large N expansion of ${\mathrm {Wg}^{\mathbf {S}}}(\sigma )$ in terms of monotone factorizations via the equation

The discussion above motivates the following definition and theorem, whose proof is immediate after setting $t = 1 - \frac {1}{q}$ in the previous equation.

Theorem 2.12 (Large N expansion)

For a permutation $\sigma \in S_k$ , we have the following large N expansion for fixed $t = 1 - \frac {N}{M}$ .

(2.3) $$ \begin{align} {\mathrm{Wg}^{\mathbf{S}}}(\sigma) = \frac{1}{(1-t)^k} \sum_{r=0}^\infty \vec{w}^{t}_{r}(\sigma) \left( -\frac{1}{N} \right)^r. \end{align} $$

2.3 Representation-theoretic interpretation

The unitary invariance of the Haar measure implies that the Weingarten function ${\mathrm {Wg}^{\mathbf {S}}}$ is a function of permutations that is constant on conjugacy classes. It is natural to ask whether it admits a natural description in the class algebra of the symmetric group. Such a description of the unitary Weingarten function ${\mathrm {Wg}^{\mathbf {U}}}$ is well understood and given in terms of the Jucys–Murphy elements in the symmetric group algebra [Reference Matsumoto and Novak40, Reference Novak44]. We will show that the Weingarten function ${\mathrm {Wg}^{\mathbf {S}}}$ can be expressed similarly.

The Jucys–Murphy elements $J_1, J_2, \ldots , J_k$ are elements of the symmetric group algebra defined by

$$\begin{align*}J_i = (1~i) + (2~i) + \cdots + (i-1~i) \in \mathbb{C}[S_k], \end{align*}$$

where we interpret the formula for $i = 1$ as $J_1 = 0$ . They were introduced independently by Jucys [Reference Jucys36] and Murphy [Reference Murphy41] and their seemingly simple definition belies their remarkable properties. For example, they commute with each other and indeed, generate a maximal commutative subalgebra of $\mathbb {C}[S_k]$ . Any symmetric function of the Jucys–Murphy elements lies in the class algebra $Z\mathbb {C}[S_k]$ and the class expansions of such expressions are of significant interest, appearing in various contexts [Reference Féray30]. Furthermore, the Jucys–Murphy elements are essential elements of the Okounkov–Vershik approach to the representation theory of symmetric groups [Reference Okounkov and Vershik45]. We will subsequently require the following results that date back to the seminal work of Jucys.

It was shown by Novak [Reference Novak44] that the unitary Weingarten function can be naturally expressed in terms of the Jucys–Murphy elements via the formula

$$\begin{align*}\sum_{\sigma \in S_k} {\mathrm{Wg}^{\mathbf{U}}}(\sigma) \, \sigma = \prod_{i = 1}^k \dfrac{1}{N+J_i}. \end{align*}$$

By considering the right side as a series in $-\frac {1}{N}$ , one observes that the coefficients are homogeneous symmetric functions of the Jucys–Murphy elements. This leads directly to the notion of monotone Hurwitz numbers, which count monotone factorizations of a given permutation with a prescribed length. The analog of the equation above for the Weingarten function ${\mathrm {Wg}^{\mathbf {S}}}$ is the following, which leads to a deformation of the monotone Hurwitz numbers, which are now weighted counts of monotone factorizations with weight equal to t to the power of the hive number. This idea was already briefly introduced in Definition 2.11, but will be studied in further detail in Section 3.

Proposition 2.14 For each positive integer k, we have the following equality in $Z\mathbb {C}[S_k]$ , the centre of the symmetric group algebra.

$$\begin{align*}\sum_{\sigma \in S_k} {\mathrm{Wg}^{\mathbf{S}}}(\sigma) \, \sigma = \prod_{i = 1}^k \dfrac{M+J_i}{N+J_i}. \end{align*}$$

Proof This is essentially an algebraic reformulation of the orthogonality relations of Theorem 2.4 and we will prove it by induction on k. The base case $k=1$ corresponds to the fact that ${\mathrm {Wg}^{\mathbf {S}}}((1)) = \frac {M}{N}$ , which is an immediate consequence of the orthogonality relation for $\sigma = (1)$ .

It remains to show that, for $k \geqslant 2$ ,

$$\begin{align*}\sum_{\sigma\in S_k} {\mathrm{Wg}^{\mathbf{S}}}(\sigma) \, \sigma =\frac{M+J_k}{N+J_k} \sum_{\sigma\in S_{k-1}} {\mathrm{Wg}^{\mathbf{S}}}(\sigma) \, \sigma^{\uparrow}, \end{align*}$$

where for any permutation $\sigma \in S_{k-1}$ , we write $\sigma ^\uparrow \in S_k$ to denote the permutation that agrees with $\sigma $ on the set $\{1, 2, \ldots , k-1\}$ and fixes k. Consider multiplying both sides of this equation by $N+J_k$ and use the definition of the Jucys–Murphy element $J_k$ to show that

(2.4) $$ \begin{align} (N + J_k) \sum_{\sigma \in S_k} {\mathrm{Wg}^{\mathbf{S}}}(\sigma) \, \sigma &= \sum_{\sigma \in S_k} \bigg( N{\mathrm{Wg}^{\mathbf{S}}}(\sigma) + \sum_{i=1}^{k-1} {\mathrm{Wg}^{\mathbf{S}}}((i ~ k) \circ \sigma) \bigg) \sigma, \end{align} $$

(2.5) $$ \begin{align} (M+J_k) \sum_{\sigma \in S_{k-1}} {\mathrm{Wg}^{\mathbf{S}}}(\sigma) \, \sigma^{\uparrow} &= \sum_{\sigma \in S_k}\bigg( \delta_{\sigma(k),k} \, M {\mathrm{Wg}^{\mathbf{S}}}(\sigma^\downarrow) \end{align} $$

(2.6) $$ \begin{align} & \qquad \qquad~~ + \sum_{i=1}^{k-1} \delta_{\sigma(i),k} {\mathrm{Wg}^{\mathbf{S}}}( [\sigma \circ (i ~ k)]^\downarrow \bigg) \sigma. \end{align} $$

The second equality is more subtle than the first and relies on the observation that for $1 \leqslant i \leqslant k-1$ ,

$$ \begin{align*} \sum_{\sigma\in S_{k-1}} {\mathrm{Wg}^{\mathbf{S}}}(\sigma) \, (i ~ k) \circ \sigma^{\uparrow} &= \sum_{\sigma \in S_k : \sigma(k) = k} {\mathrm{Wg}^{\mathbf{S}}}(\sigma^\downarrow) \, (i ~ k) \circ \sigma \\ &= \sum_{\sigma \in S_k : \sigma(i) = k} {\mathrm{Wg}^{\mathbf{S}}}([ \sigma\circ (i ~ k)]^\downarrow) \, \sigma. \end{align*} $$

Finally, the desired equality between equations (2.4) and (2.6) follows directly from the orthogonality relations of Theorem 2.4, which completes the proof.

Proposition 2.14 implies the following character formula for the Weingarten function.

Corollary 2.15 (Character formula)

Consider a permutation $\sigma \in S_k$ and let $\mathrm {id} \in S_k$ be the identity. Then

$$\begin{align*}{\mathrm{Wg}^{\mathbf{S}}}(\sigma) = \frac{1}{k!} \sum_{\lambda \vdash k} \chi^\lambda(\mathrm{id}) \, \chi^\lambda(\sigma) \prod_{\square \in \lambda} \frac{M+c(\square)}{N+c(\square)}, \end{align*}$$

where $c(\square )$ denotes the content of the box $\square $ in a Young diagram of a partition. Here and throughout, we use the notation $\lambda \vdash k$ to denote that $\lambda $ is a partition of k.

Proof By the orthogonality of characters, the identity in $Z\mathbb {C}[S_k]$ can be expressed as $e = \frac {1}{k!} \sum _{\lambda \vdash k} \chi ^\lambda (\mathrm {id}) \, \chi ^\lambda $ . Use part (b) of Proposition 2.13 to write

$$\begin{align*}\prod_{i=1}^k \frac{M+J_i}{N+J_i} \cdot e = \frac{1}{k!} \sum_{\lambda\vdash k} \left( \prod_{\square \in \lambda} \dfrac{M + c(\square)}{N+c(\square)} \right) \chi^{\lambda}(\mathrm{id}) \, \chi^\lambda. \end{align*}$$

By Proposition 2.14, ${\mathrm {Wg}^{\mathbf {S}}}(\sigma )$ is the coefficient of $\sigma $ in the above expression, and the result follows.

In summary, integrals on the space $\mathbf {S}(M,N)$ of matrices admit a convolution formula involving the Weingarten function ${\mathrm {Wg}^{\mathbf {S}}}$ . The Weingarten function is in turn determined by orthogonality relations, from which it follows that its values have a representation-theoretic interpretation in terms of Jucys–Murphy elements. An alternative perspective shows that the large N expansion of the Weingarten function for fixed $t = 1 - \frac {N}{M}$ has coefficients that are weighted counts of monotone factorisations. In the next section, we investigate the combinatorics of these enumerations in further detail.

3.1 Deformed monotone Hurwitz numbers

Monotone Hurwitz numbers enumerate monotone factorizations of permutations and arise as coefficients in the large N expansion of the unitary Weingarten function [Reference Goulden, Guay-Paquet and Novak33]. Weingarten calculus on the space $\mathbf {S}(M,N)$ naturally motivates a deformation of the monotone Hurwitz numbers, obtained by counting with a weight equal to t to the power of the hive number. This construction produces a family of polynomials with remarkable properties.

Recall from Definition 2.11 that $\vec {w}^{t}_{r}(\sigma )$ denotes the weighted count of monotone factorizations of $\sigma $ , while $\vec {h}^{t}_{r}(\sigma )$ restricts the count to transitive monotone factorizations. To fit the usual nomenclature concerning Hurwitz-type problems, we introduce a new notation to reindex by “topology,” consider the enumeration as a function on cycle type, and normalise by the product of cycle lengths.

Definition 3.1 (Deformed monotone Hurwitz numbers)

For $\mu _1, \ldots , \mu _n \geqslant 1$ , let $\sigma $ be any permutation with n cycles of lengths $\mu _1, \ldots , \mu _n$ and define

$$ \begin{align*} \vec{W}^t_{g,n}(\mu_1, \ldots, \mu_n) &= \frac{1}{\mu_1 \cdots \mu_n} \, \vec{w}^{t}_{|\mu|+2g-2+n}(\sigma), \\ \vec{H}^t_{g,n}(\mu_1, \ldots, \mu_n) &= \frac{1}{\mu_1 \cdots \mu_n} \, \vec{h}^{t}_{|\mu|+2g-2+n}(\sigma). \end{align*} $$

Here and throughout, we use the notation $|\mu |$ as a shorthand for the sum $\mu _1 + \cdots + \mu _n$ . We refer to $\vec {W}^t_{g,n}(\mu _1, \ldots , \mu _n)$ as a disconnected deformed monotone Hurwitz number and $ \vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n)$ as a connected deformed monotone Hurwitz number.

The character formula of Corollary 2.15 translates into one for deformed monotone Hurwitz numbers as follows.

Proposition 3.3 (Character formula)

Let $[\hbar ^r] F(\hbar )$ denote the coefficient of $\hbar ^r$ in the series for $F(\hbar )$ at $\hbar = 0$ and let $\chi ^\lambda _\mu $ denote the value of the symmetric group character $\chi ^\lambda $ on a permutation of cycle type $\mu $ . Then

$$\begin{align*}\vec{W}^t_{g,n}(\mu_1, \ldots, \mu_n) = \frac{1}{\mu_1 \cdots \mu_n} \, \big[ \hbar^{|\mu|+2g-2+n} \big] \sum_{\lambda \vdash |\mu|} \frac{\chi^\lambda_{11 \cdots 1} \, \chi^\lambda_\mu}{|\mu|!} \prod_{\square \in \lambda} \frac{1 - \hbar \, c(\square) + t \hbar \, c(\square)}{1 - \hbar \, c(\square)}. \end{align*}$$

Proof Equate the expressions for ${\mathrm {Wg}^{\mathbf {S}}}(\sigma )$ in terms of monotone factorizations (Theorem 2.12) and characters of the symmetric group (Corollary 2.15), using the notation $\hbar = -\frac {1}{N}$ and $t = 1 - \frac {N}{M}$ .

$$\begin{align*}\frac{1}{(1-t)^k} \sum_{r=0}^\infty \vec{w}^{t}_{r}(\sigma) \, \hbar^r = \frac{1}{k!} \sum_{\substack{\lambda\vdash k}} \chi^\lambda(\mathrm{id}) \, \chi^\lambda(\sigma) \prod_{\square \in \lambda} \frac{1 - \hbar \, c(\square) + t \hbar \, c(\square)}{(1-t) (1 - \hbar \, c(\square))}. \end{align*}$$

The desired result is obtained by multiplying through by $(1-t)^k$ , extracting the coefficient of $\hbar ^{|\mu |+2g-2+n}$ from both sides, and using Definition 3.1 for $\vec {W}^t_{g,n}(\mu _1, \ldots , \mu _n)$ .

The family of deformed monotone Hurwitz numbers is stored in the large N expansion of the following matrix integral, which we interpret as the partition function for the enumeration. As usual, the partition function naturally stores the disconnected enumeration, while its logarithm stores the connected enumeration.

Proposition 3.4 (Matrix integral)

The large N expansion of the formal matrix integral below is a partition function for the deformed monotone Hurwitz numbers. Here, we use the notation $\hbar = -\frac {1}{N}$ and $p_i$ represents the ith power sum symmetric function, evaluated on the eigenvalues of the $N \times N$ complex matrix A. (It is common to see a power of $\hbar ^{2g-2+n}$ in the partition function—the extra $\hbar ^{|\mu |}$ here can be removed by rescaling.)

$$ \begin{align*} \int_{\mathbf{S}(M,N)} \exp &\frac{N \, \mathrm{tr}(AS)}{M} \, \mathrm{d} S \\ &= 1 + \sum_{g=-\infty}^{\infty} \sum_{n=1}^\infty \sum_{\mu_1, \ldots, \mu_n = 1}^\infty \vec{W}^t_{g,n}(\mu_1, \ldots, \mu_n) \, \frac{\hbar^{|\mu|+2g-2+n}}{n!} \, p_{\mu_1} \cdots p_{\mu_n} \\ &= \exp \Bigg[ \sum_{g=0}^{\infty} \sum_{n=1}^\infty \sum_{\mu_1, \ldots, \mu_n = 1}^\infty \vec{H}^t_{g,n}(\mu_1, \ldots, \mu_n) \, \frac{\hbar^{|\mu|+2g-2+n}}{n!} \, p_{\mu_1} \cdots p_{\mu_n} \Bigg]. \end{align*} $$

3.2 Cut-and-join recursion

The notion of cut-and-join analysis was originally developed for single Hurwitz numbers, before being adapted to work in the monotone case [Reference Goulden, Guay-Paquet and Novak32, Reference Goulden, Jackson and Vainshtein34]. Deformed monotone Hurwitz numbers are amenable to a similar analysis, resulting in the following recursion.

Theorem 3.7 (Cut-and-join recursion)

Apart from the initial condition $\vec {H}^t_{0,1}(1) = 1$ , the deformed monotone Hurwitz numbers satisfy

(3.1) $$ \begin{align} &\mu_1 \vec{H}^t_{g,n}(\mu_1, \mu_S) = \sum_{i=2}^n (\mu_1 + \mu_i) \, \vec{H}^t_{g,n-1}(\mu_1 + \mu_i, \mu_{S \setminus\{i\}}) \nonumber \\&\quad + (t - 1) (\mu_1-1) \, \vec{H}^t_{g,n}(\mu_1-1, \mu_S) \nonumber \\&\quad + \sum_{\alpha+\beta=\mu_1} \alpha \beta \Bigg[ \vec{H}^t_{g-1,n+1}(\alpha, \beta, \mu_S) + \sum_{\substack{g_1+g_2 = g \\ I_1 \sqcup I_2= S}} \vec{H}^t_{g_1,|I_1|+1}(\alpha, \mu_{I_1}) \, \vec{H}^t_{g_2,|I_2|+1}(\beta, \mu_{I_2}) \Bigg],\nonumber\\ \end{align} $$

where we use the notation $S = \{2, 3, \ldots , n\}$ and $\mu _I = \{\mu _{i_1}, \mu _{i_2}, \ldots , \mu _{i_k}\}$ for $I = \{i_1, i_2, \ldots , i_k\}$ .

Proof The proof is based on the case for the usual monotone Hurwitz numbers [Reference Goulden, Guay-Paquet and Novak32], with some additional care required to keep track of the deformation parameter t. Consider multiplying equation (3.1) by $\mu _2 \cdots \mu _n$ and consider the right side to be composed of four terms in the obvious way.

Fix a permutation $\sigma \in S_k$ of cycle type $(\mu _1, \ldots , \mu _n)$ and consider the quantity $\vec {h}^{t}_{r+1}(\sigma )$ for some $r \geqslant 0$ . Note that a transitive monotone factorization

of $\sigma $ must have $\tau _{r+1} = (a~k)$ for some $1\leqslant a \leqslant k-1$ . So a transitive monotone factorization

of $\sigma $ with length $r+1$ is equivalent to a monotone factorization

of $\sigma \circ (a~k)$ with length r for some $1 \leqslant a \leqslant k-1$ , where

is transitive. This can be expressed via the equation

where we introduce the boldface notation

to denote the set $\{b_1, b_2, \ldots , b_r\}$ , excluding multiplicities, for

.

The sum on the right side can be broken down into further sums, depending on various cases. For a fixed value of a, either a is in the same cycle of $\sigma $ as k, or a is in a different cycle. The transposition $(a~k)$ is referred to as a cut or a join of $\sigma $ depending on these two cases, respectively. Moreover, if $(a~k)$ is a join of $\sigma $ , then any monotone factorization of $\sigma \circ (a~k)$ , with transitive, must be transitive itself. Based on these considerations, each term in the equation above falls into one of three cases: for each $1\leqslant a\leqslant k-1$ and with transitive, either

• • $(a~k)$ is a join of $\sigma $ and is transitive;

• • $(a~k)$ is a cut of $\sigma $ and is transitive; or

• • $(a~k)$ is a cut of $\sigma $ and is not transitive.

The first two cases can be expressed as one term to give the equation

(3.2)

In the usual notation for deformed monotone Hurwitz numbers, the first term here precisely gives rise to the first and third terms of equation (3.1), while the second term here almost gives rise to the fourth term of equation (3.1). The remainder of the proof explains the meaning of “almost” here and how the second term of equation (3.1) accounts correctly for this anomaly.

If falls into the third case for some value of $1 \leqslant a \leqslant k-1$ , then the transitive orbit of is split into two orbits of — one containing a and one containing k. In fact, these orbits are given by a partition of the disjoint cycles of $\sigma \circ (a~k)$ into two classes, thus giving rise to the quadratic terms in equation (3.1). The only instance that and differ is when k is in an orbit of size $1$ under , which then requires an additional weight of t to be contributed. The $t-1$ factor in the second term of equation (3.1) precisely handles this adjustment, since we wish to remove one of the $\vec {H}^t_{0,1}(1) \, \vec {H}^t_{g,n}(\mu _1 -1, \vec {\mu }_S)$ contributions appearing in the fourth term and weight it by an additional factor of t.

Thus, in the correct final accounting of all terms, we see that the first term of equation (3.2) corresponds to the first and third terms of equation (3.1), while the second term of equation (3.2) corresponds to the second and fourth terms of equation (3.1).

The cut-and-join recursion provides an effective algorithm for calculating deformed monotone Hurwitz numbers, which is more efficient than naive approaches. With this improved computational power comes the ability to make large-scale empirical observations on the structure of the deformed monotone Hurwitz numbers and a recursive means to prove them. The most evident of these observations are the symmetric and unimodal nature of the coefficients of $\vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n)$ .

Adopting the terminology of Brenti [Reference Brenti7] and Zeilberger [Reference Zeilberger50], we define the following.

Proof We use strong induction on the value of $r = |\mu | + 2g - 2 + n \geqslant 1$ . The only deformed monotone Hurwitz number corresponding to $r = 1$ is $\vec {H}^t_{0,1}(2) = \frac {1}{2} t$ , which is indeed a $\Lambda $ -polynomial of degree $1$ and darga $2$ . Now consider a deformed monotone Hurwitz number $\vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n)$ corresponding to $r \geqslant 2$ and rewrite the cut-and-join recursion of Theorem 3.7 as

$$ \begin{align*} &\mu_1 \vec{H}^t_{g,n}(\mu_1, \mu_S) = \sum_{i=2}^n (\mu_1 + \mu_i) \, \vec{H}^t_{g,n-1}(\mu_1 + \mu_i, \mu_{S \setminus\{i\}}) \\&\quad + (t + 1) (\mu_1-1) \, \vec{H}^t_{g,n}(\mu_1-1, \mu_S) \\&\quad + \sum_{\alpha+\beta=\mu_1} \alpha \beta \Bigg[ \vec{H}^t_{g-1,n+1}(\alpha, \beta, \mu_S) + \sum_{\substack{g_1+g_2 = g \\ I_1 \sqcup I_2= S}}^{\text{no } \vec{H}^t_{0,1}(1)} \vec{H}^t_{g_1,|I_1|+1}(\alpha, \mu_{I_1}) \, \vec{H}^t_{g_2,|I_2|+1}(\beta, \mu_{I_2}) \Bigg]. \end{align*} $$

Here, we have simply removed the two terms including $\vec {H}^t_{0,1}(1)$ from the final summation and incorporated them into the second term on the right side.

By the inductive hypothesis, all deformed monotone Hurwitz numbers appearing on the right side of the cut-and-join recursion are $\Lambda $ -polynomials. To prove that $\vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n)$ is also one, we rely on the following closure properties of $\Lambda $ -polynomials, proofs of which can be found in [Reference Sun, Wang and Zhang47].

• • If P and Q are $\Lambda $ -polynomials, then the product $PQ$ is a $\Lambda $ -polynomial with $\mathrm {dar}(PQ) = \mathrm {dar}(P) + \mathrm {dar}(Q)$ .

• • If P and Q are $\Lambda $ -polynomials of the same darga d, then $P + Q$ is a $\Lambda $ -polynomial of darga d.

It follows that each of the four terms on the right side of the equation above are $\Lambda $ -polynomials of darga $|\mu |$ , or possibly equal to the zero polynomial. So the entire right side, and hence $\vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n)$ , is a $\Lambda $ -polynomial of darga $|\mu |$ .

For any permutation $\sigma \in S_k$ for $k \geqslant 2$ , one can construct a transitive monotone factorization of $\sigma $ with hive number $1$ . So the polynomial $\vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n)$ has a non-zero linear term, but no constant term. Combined with the fact that it is a $\Lambda $ -polynomial of darga $|\mu |$ , it follows that $\vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n)$ is a polynomial of degree $|\mu |-1$ .

As we will see in Section 4, deformed monotone Hurwitz numbers belong to a large class of enumerative problems governed by the topological recursion. In many such instances, the one-point invariants—in other words, those with $n = 1$ —are governed by a recursion that is linear, rather than quadratic like the cut-and-join recursion. In previous work of Chaudhuri and the second author [Reference Chaudhuri and Do9], it was shown that such one-point recursions exist for “weighted Hurwitz numbers,” to use the terminology of Alexandrov, Chapuy, Eynard, and Harnad [Reference Alexandrov, Chapuy, Eynard and Harnad1]. Furthermore, they gave an explicit algorithmic approach to obtain these recursions that is effective in many examples. In the case of deformed monotone Hurwitz numbers, one obtains the following result.

Theorem 3.12 (One-point recursion)

The one-point deformed monotone Hurwitz numbers—in other words, those with $n = 1$ —satisfy

$$ \begin{align*} d^2 \, \vec{H}^t_{g,1}(d) ={}& (d-1) (2d-3) (t+1) \, \vec{H}^t_{g,1}(d-1) \\ &- (d-2) (d-3) (t-1)^2 \, \vec{H}^t_{g,1}(d-2) + d^2(d-1)^2 \, \vec{H}^t_{g-1,1}(d). \end{align*} $$

Observe that setting $t = 1$ above recovers the known one-point recursion for monotone Hurwitz numbers [Reference Chaudhuri and Do9]. Furthermore, setting $g = 0$ and combining with equation (1.1) recovers the known linear recursion for Narayana polynomials [Reference Fisk31].

3.3 Interlacing phenomena

Root conjectures

The cut-and-join recursion of Theorem 3.7 can be used to effectively compute a large number of deformed monotone Hurwitz numbers, some of which are shown in Appendix A. We previously stated in Proposition 3.9 that the coefficients of these polynomials are symmetric and unimodal. We now turn our attention to their roots, which exhibit rather striking behavior that remains largely conjectural at present. The concerning roots of deformed monotone Hurwitz numbers is the following.

Conjecture 3.13 (Real-rootedness)

For all $g \geqslant 0$ , $n \geqslant 1$ and $\mu _1, \ldots , \mu _n \geqslant 1$ , the deformed monotone Hurwitz number $\vec {H}^t_{g,n}(\mu _1, \mu _2, \ldots , \mu _n)$ is a real-rooted polynomial in t.

The roots of the deformed monotone Hurwitz numbers are not only real, but also possess interesting structure relative to each other. We say that a polynomial P interlaces a polynomial Q if

• • the degree of P is n and the degree of Q is $n+1$ for some positive integer n;

• • P has n real roots $a_1 \leqslant a_2 \leqslant \cdots \leqslant a_n$ and Q has $n+1$ real roots $b_1 \leqslant b_2 \leqslant \cdots \leqslant b_{n+1}$ , allowing for multiplicity; and

• • $b_1 \leqslant a_1 \leqslant b_2 \leqslant a_2 \leqslant \cdots \leqslant b_n \leqslant a_n \leqslant b_{n+1}$ .

If the inequalities above are strict, then we say that the polynomial P strictly interlaces the polynomial Q. By convention, we will also say that a polynomial P (strictly) interlaces a polynomial Q if P is constant and Q is affine.

It is known that the sequence of Narayana polynomials interlace in the sense that $\mathrm {Nar}_\mu (t)$ interlaces $\mathrm {Nar}_{\mu +1}(t)$ for every positive integer $\mu $ [Reference Fisk31]. Given equation (1.1), one may wonder whether an analogous property persists more generally across the whole family of deformed monotone Hurwitz numbers. Overwhelming empirical evidence suggests that this is indeed the case and we have the following.

Conjecture 3.14 (Interlacing)

The polynomial $\vec {H}^t_{g,n}(\mu _1, \mu _2, \ldots , \mu _n)$ interlaces each of the n polynomials

$$\begin{align*}\vec{H}^t_{g,n}(\mu_1+1, \mu_2, \ldots, \mu_n), \!\quad \vec{H}^t_{g,n}(\mu_1, \mu_2+1, \ldots, \mu_n), \!\!\quad\!\! \ldots, \quad \vec{H}^t_{g,n}(\mu_1, \mu_2, \ldots, \mu_n+1). \end{align*}$$

Conjecture 3.14 states that for fixed $(g,n)$ , one obtains a lattice of interlacing polynomials. It has been checked computationally for many cases, including the following, amounting to 1,430 independent checks:

• • $g \in \{0, 1\}$ , $n \in \{1, 2, 3, 4, 5\}$ , $|\mu | \leqslant 15$ ;

• • $g \in \{2, 3\}$ , $n \in \{1, 2, 3, 4, 5\}$ , $|\mu | \leqslant 12$ ;

• • $g \in \{4, 5\}$ , $n \in \{1, 2, 3, 4, 5\}$ , $|\mu | \leqslant 10$ .

One can observe rich structure in the roots of the deformed monotone Hurwitz numbers and write down further conjectures. As an example, consider the following, which asserts that the roots of $\vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n)$ behave well for fixed $\mu _1, \ldots , \mu _n$ and increasing g. A great deal of data can be generated to support this conjecture, some examples of which appear in the tables of Figures 2 and 3.

Figure 2: The roots $\alpha _1 \leqslant \alpha _2 \leqslant \cdots \leqslant \alpha _6$ of $\vec {H}^t_{20,n}(\mu _1, \ldots , \mu _n)$ for all $\mu _1, \ldots , \mu _n$ satisfying $|\mu | = 7$ , to 9 decimal places. The roots $\alpha _3 = -1$ and $\alpha _6 = 0$ have been omitted. Observe the proximity of the numbers in each column to $-\frac {5}{1}, -\frac {4}{2}, -\frac {2}{4}, -\frac {1}{5}$ , as predicted by Conjecture 3.15.

Figure 3: The roots $\alpha _1 \leqslant \alpha _2 \leqslant \cdots \leqslant \alpha _6$ of $\vec {H}^t_{g,3}(4, 2, 1)$ for $g = 10, 11, 12, \ldots , 20$ , to 11 decimal places. The roots $\alpha _3 = -1$ and $\alpha _6 = 0$ have been omitted. Observe the monotonic convergence of the numbers in each column to $-\frac {5}{1}, -\frac {4}{2}, -\frac {2}{4}, -\frac {1}{5}$ , as predicted by Conjecture 3.15.

Conjecture 3.15 Fix positive integers $\mu _1, \ldots , \mu _n$ whose sum is d. As $g \to \infty $ , the roots of the polynomial $\vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n)$ , in order from smallest to largest, respectively approach the values

$$\begin{align*}- \frac{d-2}{1}, ~~ - \frac{d-3}{2}, ~~ - \frac{d-4}{3}, ~~ \ldots, ~~ - \frac{2}{d-3}, ~~ - \frac{1}{d-2}, ~~ 0. \end{align*}$$

Moreover, the convergence to a number less than $-1$ is increasing from below, while the convergence to a non-zero number greater than $-1$ is decreasing from above.

An interlacing result

There is a rich theory concerning interlacing polynomials, as evidenced by the tome of Fisk [Reference Fisk31]. The following lemma is a slight adaptation of [Reference Fisk31, Lemma 1.82], and will be used to prove the $(g,n) = (0,1)$ and $(g,n) = (1,1)$ cases of Conjectures 3.13 and 3.14.

Lemma 3.16 Let $f_1(t), f_2(t), \ldots $ be a sequence of polynomials with non-negative coefficients, such that $\deg f_i = i$ . Suppose that $f_1(t)$ strictly interlaces $f_2(t)$ , and that the sequence of polynomials satisfies the relation

(3.3) $$ \begin{align} f_{d+1}(t) = \ell_d(t) \, f_{d}(t) - q_d(t) \, f_{d-1}(t), \end{align} $$

where $\ell _d(t)$ is an affine function and $q_d(t)$ is a quadratic function that is positive for $t \leqslant 0$ . Then all polynomials in the sequence are real-rooted and $f_i(t)$ strictly interlaces $f_{i+1}(t)$ for every positive integer i.

Proof Suppose that $f_{d-1}(t)$ strictly interlaces $f_d(t)$ . Since $f_d(t)$ has non-negative integer coefficients, its roots can be written as $0 \geqslant r_1> r_2 > \cdots > r_d$ . Then equation (3.3) evaluated at one of these roots yields

$$\begin{align*}f_{d+1}(r_i) = - q_d(r_i) \, f_{d-1}(r_i), \end{align*}$$

which implies that $f_{d+1}(r_i)$ and $f_{d-1}(r_i)$ have different signs. Since $f_{d-1}(t)$ strictly interlaces $f_d(t)$ , its values at $r_1, r_2, \ldots , r_d$ must alternate in sign, leading to the following table. The entries indicate the signs of $f_{d-1}(t)$ and $f_{d+1}(t)$ , evaluated at $r_1, r_2, \ldots , r_d$ and in the limit $t \to \pm \infty $ .

The intermediate value theorem implies that $f_{d+1}(t)$ has $d+1$ real roots and that $f_d(t)$ strictly interlaces $f_{d+1}(t)$ . The result then follows by induction on d.

Proposition 3.17 Conjectures 3.13 and 3.14 hold for $(g,n) = (0,1)$ and $(g,n) = (1,1)$ .

Proof Setting $g = 0$ in the one-point recursion of Theorem 3.12 yields

(3.4) $$ \begin{align} d^2 \vec{H}^t_{0,1}(d) = (d-1) (2d-3) (t+1) \, \vec{H}^t_{0,1}(d-1) - (d-2) (d-3) (t-1)^2 \, \vec{H}^t_{0,1}(d-2). \end{align} $$

Divide both sides by $d^2t$ to obtain a recursion governing the polynomials $\frac {\vec {H}^t_{0,1}(1)}{t}, \frac {\vec {H}^t_{0,1}(2)}{t}, \frac {\vec {H}^t_{0,1}(3)}{t}, \ldots $ of the form of equation (3.3). It follows from Lemma 3.16 that this sequence is real-rooted and strictly interlacing. Therefore, the sequence $\vec {H}^t_{0,1}(1), \vec {H}^t_{0,1}(2), \vec {H}^t_{0,1}(3), \ldots $ is a sequence of real-rooted polynomials in which each polynomial interlaces the next.

For the genus 1 case, we argue in exactly the same way, using the as yet unproven claim that

(3.5) $$ \begin{align} d(d-2) \, \vec{H}^t_{1,1}(d) = (d-1) (2d-1) (t+1) \, \vec{H}^t_{1,1}(d-1) - (d-2) (d+1) (t-1)^2 \, \vec{H}^t_{1,1}(d-2). \end{align} $$

It remains to prove this relation, which we do using a generating function approach. Multiply both sides by $x^{d-1}$ and sum over all positive integers d. Setting $w_{1,1}(x) = \sum _{d=1}^\infty d \, \vec {H}^t_{1,1}(d) \, x^{d-1}$ , the claim is equivalent to

(3.6) $$ \begin{align} -x \left[ (t-1)^2 x^2 - 2(t+1) x + 1 \right] \frac{\partial}{\partial x} w_{1,1}(x) = \left[ 4(t-1)^2 x^2 - 3(t+1) x - 1 \right] w_{1,1}(x). \end{align} $$

We borrow Theorem 4.1 from the next section, which allows us to explicitly compute via $w_{1,1}(x) = \frac {\omega _{1,1}(z)}{\mathrm {d} x}$ that

(3.7) $$ \begin{align} w_{1,1}(x) = -\frac{t z (z-1) (1 - z + tz)^4}{(tz^2 - z^2 + 2z - 1)^5}, \end{align} $$

where the coordinates x and z are related by $x = \frac {z(1 - z)}{1-z+tz}$ . One can now check that the expression for $w_{1,1}(x)$ of equation (3.7) satisfies the differential equation of equation (3.6) using a computer algebra system, thereby proving equation (3.5).

Remark 3.18. It is natural to seek higher genus analogs of equations (3.4) and (3.5), in order to extend the result and proof of Proposition 3.17. However, one can prove that there exists no recursion of the form

$$\begin{align*}d A(d) \, \vec{H}^t_{2,1}(d) = (d-1) B(d) f(t) \, \vec{H}^t_{2,1}(d-1) - (d-2) C(d) g(t) \, \vec{H}^t_{2,1}(d-2), \end{align*}$$

where $A(d), B(d), C(d)$ are affine functions of d, $f(t)$ is an affine function of t, and $g(t)$ is a quadratic function of t. So the exact argument used in the proof of Proposition 3.17 cannot extend to genus 2 and higher without some alteration.

4.1 Topological recursion for deformed monotone Hurwitz numbers

The topological recursion arose in the mathematical physics literature as a formalism to capture loop equations in matrix models [Reference Chekhov and Eynard11, Reference Eynard and Orantin25]. There is now an extensive theory around topological recursion and it is known to govern a wide range of enumerative-geometric problems beyond the original matrix model context. These include: Hurwitz numbers and various generalizations [Reference Borot, Do, Karev, Lewański and Moskovsky2, Reference Bouchard, Serrano, Liu and Mulase4, Reference Bouchard and Mariño6, Reference Do, Leigh and Norbury19, Reference Dunin-Barkowski, Kramer, Popolitov and Shadrin23, Reference Eynard, Mulase and Safnuk27], monotone Hurwitz numbers [Reference Do, Dyer and Mathews18], lattice points in moduli spaces of curves [Reference Chaudhuri, Do and Moskovsky10, Reference Norbury42], intersection theory on moduli spaces of curves [Reference Eynard24, Reference Eynard and Orantin25], Gromov–Witten theory of $\mathbb {CP}^1$ [Reference Norbury and Scott43, Reference Dunin-Barkowski, Orantin, Shadrin and Spitz22], Gromov–Witten theory of toric Calabi–Yau threefolds [Reference Bouchard, Klemm, Mariño and Pasquetti5, Reference Eynard and Orantin26, Reference Fang, Liu and Zong29], and cohomological field theories [Reference Dunin-Barkowski, Orantin, Shadrin and Spitz22]. The topological recursion is also conjectured to govern certain quantum knot invariants, such as the large N asymptotics of colored Jones polynomials [Reference Borot and Eynard3, Reference Dijkgraaf, Fuji and Manabe17] and colored HOMFLY-PT polynomials [Reference Gu, Jockers, Klemm and Soroush35].

In its original formulation due to Chekhov, Eynard, and Orantin [Reference Chekhov and Eynard11, Reference Eynard and Orantin25], the topological recursion takes as input a spectral curve $(\mathcal {C}, x, y, \omega _{0,2})$ comprising a compact Riemann surface $\mathcal {C}$ equipped with two meromorphic functions $x, y: \mathcal {C} \to \mathbb {CP}^1$ and a bidifferential $\omega _{0,2}$ with a double pole along the diagonal. These are required to satisfy mild technical assumptions, which we do not mention here. Indeed, rather than fully define the topological recursion, we refer the reader to the many existing expositions in the literature [Reference Do, Dyer and Mathews18, Reference Eynard and Orantin28]. For the present discussion, it suffices to note that the topological recursion uses the initial data to define the base cases $\omega _{0,1}(z_1) = y(z_1) \, \mathrm {d} x(z_1)$ and $\omega _{0,2}(z_1, z_2)$ , and then produces so-called correlation differentials via the following recursive formula.Footnote ³

$$ \begin{align*} \omega_{g,n}(z_1, \ldots, z_n) = \sum_{\alpha} \mathop{\mathrm{Res}}_{z = \alpha}~ &K(z_1, z) \Bigg[ \omega_{g-1,n+1}(z, \sigma_\alpha(z), z_2, \ldots, z_n) \\ &+ \sum_{\substack{g_1+g_2=g \\ I_1 \sqcup I_2 = \{2, \ldots, n\}}}^\circ \omega_{g_1,|I_1|+1}(z, z_{I_1}) \, \omega_{g_2,|I_2|+1}(\sigma_\alpha(z), z_{I_2}) \Bigg]. \end{align*} $$

In various settings, the correlation differential $\omega _{g,n}$ acts as a generating function for enumerative-geometric quantities, where g represents the genus of a surface and n its number of punctures, or boundaries.

It was previously shown that the monotone Hurwitz numbers are governed by the topological recursion in the following sense [Reference Do, Dyer and Mathews18]. Topological recursion on the spectral curveFootnote ⁴

$$\begin{align*}\mathcal{C} = \mathbb{CP}^1, \quad x(z) = z(1 - z), \quad y(z) = \frac{1}{1 - z}, \quad \omega_{0,2}(z_1, z_2) = \frac{\mathrm{d} z_1 \, \mathrm{d} z_2}{(z_1 - z_2)^2} \end{align*}$$

produces correlation differentials satisfying

$$ \begin{align*} \omega_{g,n}(z_1, \ldots, z_n) &= \mathrm{d}_1 \cdots \mathrm{d}_n \sum_{\mu_1, \ldots, \mu_n=1}^{\infty} \vec{H}_{g,n} (\mu_1, \ldots, \mu_n) \, x(z_1)^{\mu_1} \cdots x(z_n)^{\mu_n} \\&\quad + \delta_{g,0} \delta_{n,2} \, \frac{\mathrm{d} x(z_1) \, \mathrm{d} x(z_2)}{\left( x(z_1) - x(z_2) \right)^2}. \end{align*} $$

Here, $\mathrm {d}_i$ represents the exterior derivative in the ith coordinate and $\vec {H}_{g,n}(\mu _1, \ldots , \mu _n) = \big [ \vec {H}^t_{g,n}(\mu _1, \ldots , \mu _n) \big ]_{t=1}$ denotes a monotone Hurwitz number.

It is thus natural to consider whether the deformed monotone Hurwitz numbers satisfy the topological recursion on a deformation of the spectral curve above. Perhaps unsurprisingly, this is indeed the case and follows from the representation-theoretic interpretation for deformed monotone Hurwitz numbers of Corollary 2.15 and a powerful result of Bychkov, Dunin-Barkowski, Kazarian, and Shadrin [Reference Bychkov, Dunin-Barkowski, Kazarian and Shadrin8], which builds on previous work of Alexandrov, Chapuy, Eynard, and Harnad [Reference Alexandrov, Chapuy, Eynard and Harnad1].

Theorem 4.1 Topological recursion on the spectral curve

$$\begin{align*}\mathcal{C} = \mathbb{CP}^1, \quad x(z) = \frac{z(1 - z)}{1-z+tz}, \quad y(z) = \frac{1-z+tz}{1 - z}, \quad \omega_{0,2}(z_1, z_2) = \frac{\mathrm{d} z_1 \, \mathrm{d} z_2}{(z_1 - z_2)^2} \end{align*}$$

produces correlation differentials satisfying

$$ \begin{align*} \omega_{g,n}(z_1, \ldots, z_n) &= \mathrm{d}_1 \cdots \mathrm{d}_n \sum_{\mu_1, \ldots, \mu_n=1}^{\infty} \vec{H}^t_{g,n} (\mu_1, \ldots, \mu_n) \, x(z_1)^{\mu_1} \cdots x(z_n)^{\mu_n} \\&\quad + \delta_{g,0} \delta_{n,2} \, \frac{\mathrm{d} x(z_1) \, \mathrm{d} x(z_2)}{\left( x(z_1) - x(z_2) \right)^2}. \end{align*} $$

Proof The main theorem of Bychkov, Dunin-Barkowski, Kazarian, and Shadrin in [Reference Bychkov, Dunin-Barkowski, Kazarian and Shadrin8] states that the topological recursion governs the coefficients of certain KP tau functions of hypergeometric type. We present here a simplified version of their result that is fit for purpose.

Let $\tilde {y}(z) = \sum _{i=1}^\infty y_i \, z^i$ be the series expansion of a rational function satisfying $\tilde {y}(0) = 0$ and let $f(z)$ be a rational function satisfying $f(0) = 1$ . From this data, define the generating function

$$\begin{align*}Z(p_1, p_2, \ldots; \hbar) = \sum_{\lambda \in \mathcal{P}} s_\lambda(p_1, p_2, \ldots) \, s_\lambda \Big( \frac{y_1}{\hbar}, \frac{y_2}{\hbar}, \ldots \Big) \prod_{\square \in \lambda} f(\hbar \, c(\square)), \end{align*}$$

where $\mathcal {P}$ is the set of partitions, $s_\lambda $ represents the Schur function expressed in terms of power sum symmetric functions, and $c(\square )$ denotes the content of the box $\square $ in the Young diagram of a partition. This is a KP tau function of hypergeometric type and there exist “weighted Hurwitz numbers” $X_{g,n}(\mu _1, \ldots , \mu _n)$ for $g \geqslant 0$ , $n \geqslant 1$ and $\mu _1, \ldots , \mu _n \geqslant 1$ such that

(4.1) $$ \begin{align} Z(p_1, p_2, \ldots; \hbar) = \exp \bigg( \sum_{g=0}^\infty \sum_{n=1}^\infty \frac{\hbar^{2g-2+n}}{n!} \sum_{\mu_1, \ldots, \mu_n=1}^\infty X_{g,n}(\mu_1, \ldots, \mu_n) \, p_{\mu_1} \cdots p_{\mu_n} \bigg). \end{align} $$

Bychkov, Dunin-Barkowski, Kazarian, and Shadrin [Reference Bychkov, Dunin-Barkowski, Kazarian and Shadrin8] prove that topological recursion on the spectral curve

$$\begin{align*}\mathcal{C} = \mathbb{CP}^1, \quad x(z) = \frac{z}{f(\tilde{y}(z))}, \quad y(z) = \frac{\tilde{y}(z)}{x(z)}, \quad \omega_{0,2}(z_1, z_2) = \frac{\mathrm{d} z_1 \, \mathrm{d} z_2}{(z_1 - z_2)^2} \end{align*}$$

produces correlation differentials satisfying

$$ \begin{align*} \omega_{g,n}(z_1, \ldots, z_n) &= \mathrm{d}_1 \cdots \mathrm{d}_n \sum_{\mu_1, \ldots, \mu_n=1}^{\infty} X_{g,n}\ (\mu_1, \ldots, \mu_n) \, x(z_1)^{\mu_1} \cdots x(z_n)^{\mu_n} \\&\quad + \delta_{g,0} \delta_{n,2} \, \frac{\mathrm{d} x(z_1) \, \mathrm{d} x(z_2)}{\left( x(z_1) - x(z_2) \right)^2}. \end{align*} $$

We simply specialise this result to $\tilde {y}(z) = z$ and $f(z) = \frac {1-z+tz}{1-z}$ . It remains to check that the monotone deformed Hurwitz numbers agree with the weighted Hurwitz numbers with this particular choice of data.

The exponential of equation (4.1) transforms the connected generating function to the disconnected generating function, so it suffices to show that the disconnected deformed monotone Hurwitz numbers satisfy

$$\begin{align*}\vec{W}^t_{g,n}(\mu_1, \ldots, \mu_n) = |\mathrm{Aut}(\mu)| \, \big[ \hbar^{2g-2+n} p_{\mu_1} \cdots p_{\mu_n} \big] Z(p_1, p_2, \ldots; \hbar). \end{align*}$$

Here, $\mathrm {Aut}(\mu )$ denotes the subgroup of permutations in $S_n$ that fix $(\mu _1, \ldots , \mu _n)$ . This automorphism factor arises because the summation in equation (4.1) is over all tuples of positive integers, rather than partitions.

Now use $s_\lambda (p_1, p_2, \ldots ) = \displaystyle \sum _{\mu \vdash |\lambda |} \frac {\chi ^\lambda _\mu }{|\mathrm {Aut}(\mu )|} \prod \frac {p_{\mu _i}}{\mu _i}$ in the expression

$$\begin{align*}Z(p_1, p_2, \ldots; \hbar) = \sum_{\lambda \in \mathcal{P}} s_\lambda(p_1, p_2, \ldots) \, s_\lambda ( \tfrac{1}{\hbar}, 0, 0, \ldots ) \prod_{\square \in \lambda} \frac{1 - \hbar \, c(\square) + t \hbar \, c(\square)}{1 - \hbar \, c(\square)} \end{align*}$$

to obtain

$$ \begin{align*} |\mathrm{Aut}(\mu)| \, &\big[ \hbar^{2g-2+n} p_{\mu_1} \cdots p_{\mu_n} \big] Z(p_1, p_2, \ldots; \hbar) \\ &\qquad = \frac{1}{\mu_1 \cdots \mu_n} \, \big[ \hbar^{|\mu|+2g-2+n} \big] \sum_{\lambda \vdash |\mu|} \frac{\chi^\lambda_{11 \cdots 1} \, \chi^\lambda_\mu}{|\mu|!} \prod_{\square \in \lambda} \frac{1 - \hbar \, c(\square) + t \hbar \, c(\square)}{1 - \hbar \, c(\square)}. \end{align*} $$

That this is equal to $\vec {W}^t_{g,n}(\mu _1, \ldots , \mu _n)$ is precisely the content of Proposition 3.3, and this concludes the proof.

Remark 4.2. The meromorphic functions $x, y: \mathbb {CP}^1 \to \mathbb {CP}^1$ of Theorem 4.1 satisfy

$$\begin{align*}xy^2 + (t-1)xy - y + 1 = 0, \end{align*}$$

which is the global form of the spectral curve. The general theory of topological recursion asserts that, for $(g,n) \neq (0,1)$ or $(0,2)$ , the correlation differential $\omega _{g,n}$ is a rational multidifferential on this curve with poles only at the zeroes of $\mathrm {d} x$ and further conditions on the pole structure [Reference Eynard and Orantin25]. This in turn leads to a polynomiality structure theorem for the deformed monotone Hurwitz numbers. Furthermore, the relation between topological recursion and cohomological field theories should lead to an interpretation of deformed monotone Hurwitz numbers as intersection numbers on moduli spaces of curves [Reference Dunin-Barkowski, Orantin, Shadrin and Spitz22, Reference Eynard24]. However, we do not pursue this investigation further in the present work.

4.2 Topologising sequences of polynomials

From Catalan curves to Narayana curves

As previously mentioned in equation (1.1), the $(g,n) = (0,1)$ case of the deformed monotone Hurwitz numbers recovers the Narayana polynomials in the following way.

$$\begin{align*}(\mu+1) \, \vec{H}^t_{0,1}(\mu+1) = \mathrm{Nar}_{\mu}(t) := \sum_{i=1}^{\mu} \frac{1}{\mu} \binom{\mu}{i} \binom{\mu}{i-1} \, t^i. \end{align*}$$

Hence, we consider the deformed monotone Hurwitz numbers to be a “topological generalization” of the Narayana polynomials. This viewpoint appears throughout the literature, such as in Dumitrescu and Mulase’s exposition on generalized Catalan numbers [Reference Dumitrescu and Mulase21].

We propose that the topological recursion can be used as a mechanism to “topologise” sequences of polynomials. One would apply topological recursion to a spectral curve with a deformation parameter t to obtain a family of polynomials $X_{g,n}^t(\mu _1, \ldots , \mu _n)$ for $g \geqslant 0$ , $n \geqslant 1$ and $\mu _1, \ldots , \mu _n \geqslant 1$ such that $X^t_{0,1}(1), X^t_{0,1}(2), \ldots $ recovers the original sequence of polynomials in t. Furthermore, we claim that interesting properties of the original sequence of polynomials can persist into the topological generalization. Examples of such properties are the symmetry and unimodality of coefficients as well as the interlacing of roots. These both hold for Narayana polynomials and are respectively proven (Proposition 3.9) and conjectured (Conjecture 3.14) to hold for deformed monotone Hurwitz numbers more generally.

There is more than one way to topologise a given sequence of polynomials. In order to obtain a topological generalization of the Narayana numbers, it is natural to consider spectral curves that store Catalan numbers in the correlation differential $\omega _{0,1}$ and to consider a suitable t-deformation. The literature suggests three natural candidates for such “Catalan spectral curves” and these curves govern monotone Hurwitz numbers, map enumeration, and dessin d’enfant enumeration. This leads to the following three “Narayana spectral curves” with $\mathcal {C} = \mathbb {CP}^1$ , $\omega _{0,2} = \frac {\mathrm {d} z_1 \, \mathrm {d} z_2}{(z_1 - z_2)^2}$ , and the meromorphic functions $x, y: \mathcal {C} \to \mathbb {CP}^1$ satisfying the following.

• • Monotone Hurwitz numbers. We have

  $$\begin{align*}y(x) = \sum_{\mu=0}^\infty \mathrm{Cat}_\mu \, x^\mu \quad \Rightarrow \quad x y^2 - y + 1 = 0, \end{align*}$$
  which is the global form of the spectral curve for monotone Hurwitz numbers [Reference Do, Dyer and Mathews18]. Promoting the Catalan numbers to the Narayana polynomials, one obtains
  $$\begin{align*}y(x) = \sum_{\mu=0}^\infty \mathrm{Nar}_\mu(t) \, x^\mu \quad \Rightarrow \quad xy^2 + (t-1)xy - y + 1 = 0. \end{align*}$$
  This is the global form of the spectral curve for deformed monotone Hurwitz numbers, which appeared above in Theorem 4.1.

• • Map enumeration. We have

  $$\begin{align*}y(x) = \sum_{\mu=0}^\infty \mathrm{Cat}_\mu \, x^{-2\mu-1} \quad \Rightarrow \quad y^2 - xy + 1 = 0, \end{align*}$$
  which is the global form of the spectral curve for the enumeration of maps [Reference Chekhov and Eynard11, Reference Eynard and Orantin25]. Promoting the Catalan numbers to the Narayana polynomials, one obtains
  $$\begin{align*}y(x) = \sum_{\mu=0}^\infty \mathrm{Nar}_\mu(t) \, x^{-2\mu-1} \quad \Rightarrow \quad xy^2 - x^2y + (t-1) y + x = 0. \end{align*}$$
  To the best of our knowledge, this is the global form of a spectral curve that has not yet been studied in the literature. We do not investigate it further in the present work, largely due to the fact that it has genus 1 for generic values of t. The topological recursion on spectral curves of positive genus is much less straightforward than on those of genus 0. Nevertheless, it would be interesting to understand this spectral curve and whether it governs a natural enumerative problem.

• • Dessin d’enfant enumeration. We have

  $$\begin{align*}y(x) = \sum_{\mu=0}^\infty \mathrm{Cat}_\mu \, x^{-\mu-1} \quad \Rightarrow \quad x y^2 - x y + 1 = 0, \end{align*}$$
  which is the global form of the spectral curve for the enumeration of dessins d’enfant [Reference Do and Norbury20, Reference Kazarian and Zograf37]. Promoting the Catalan numbers to the Narayana polynomials, one obtains
  $$\begin{align*}y(x) = \sum_{\mu=0}^\infty \mathrm{Nar}_\mu(t) \, x^{-\mu-1} \quad \Rightarrow \quad x y^2 - x y + (t-1)y + 1 = 0. \end{align*}$$
  This is the global form of a spectral curve with genus 0. It leads to a topological generalisation of the Narayana polynomials that differs from the deformed monotone Hurwitz numbers. We will consider its properties in the next section.

Remark 4.3. We certainly do not claim that the spectral curves above provide the only natural topological generalizations of the Catalan numbers or the Narayana polynomials. Indeed, one could study spectral curves arising from $y(x) = \sum _{\mu =0}^\infty \mathrm {Cat}_\mu \, x^{a\mu +b}$ or $y(x) = \sum _{\mu =0}^\infty \mathrm {Nar}_\mu (t) \, x^{a\mu +b}$ for judicious choices of a and b, and it is not unlikely that these would lead to interesting enumerative problems.

Deforming the dessin d’enfant enumeration

The previous discussion motivates the study of the spectral curve whose global form is $x y^2 - x y + (t-1)y + 1 = 0$ . In parametrized form, this corresponds to the spectral curve

$$\begin{align*}\mathcal{C} = \mathbb{CP}^1, \quad x(z) = \frac{1-z+tz}{z(1-z)}, \quad y(z) = z, \quad \omega_{0,2}(z_1, z_2) = \frac{\mathrm{d} z_1 \, \mathrm{d} z_2}{(z_1 - z_2)^2}. \end{align*}$$

By construction, this spectral curve should govern a t-deformation of the usual dessin d’enfant enumeration. In the following, we prove that this enumeration arises by attaching a multiplicative weight t to each black vertex of a dessin d’enfant. Moreover, we observe that the polynomials arising from this construction empirically satisfy real-rootedness and interlacing properties analogous to those observed for deformed monotone Hurwitz numbers in Conjectures 3.13 and 3.14. Although the weighted enumeration of dessins d’enfant has been studied previously [Reference Kazarian and Zograf37], these real-rootedness and interlacing properties have not been observed in the literature, to the best of our knowledge. Our discussion of the weighted enumeration of dessins d’enfant should be considered as a further case study promoting the viewpoint that topological recursion produces interesting topological generalizations of sequences of polynomials.

First, let us define dessins d’enfant—in other words, bicolored maps—and their enumeration.

Definition 4.4 A map is a finite graph—possibly with loops or multiple edges—embedded in a compact oriented surface such that the complement of the graph is a disjoint union of topological disks. We refer to these disks as faces and require that they are labeled $1, 2, 3, \ldots , n$ , where n denotes the number of faces.

A dessin d’enfant is a map whose vertices have been colored black or white such that each edge is incident to one black vertex and one white vertex. The degree of a face is defined to be half the number of edges incident to it.

An equivalence between two maps (or dessins d’enfant) is a bijection between their respective vertices, edges and faces that is realised by an orientation-preserving homeomorphism of their underlying surfaces and preserves all adjacencies, incidences, labels (and colors). An automorphism of a map (or dessin d’enfant) is an equivalence from the map (or dessin d’enfant) to itself.

Definition 4.5 For $g \geqslant 0$ , $n \geqslant 1$ and $\mu _1, \ldots , \mu _n \geqslant 1$ , let $D^t_{g,n}(\mu _1, \ldots , \mu _n) \in \mathbb {Q}[t]$ denote the weighted enumeration of connected genus g dessins d’enfant with n labeled faces of degrees $\mu _1, \ldots , \mu _n$ . The weight attached to a dessin d’enfant $\Gamma $ is $\frac {t^{b(\Gamma )}}{|\mathrm {Aut}(\Gamma )|}$ , where $b(\Gamma )$ denotes the number of black vertices of $\Gamma $ and $\mathrm {Aut}(\Gamma )$ denotes the automorphism group of $\Gamma $ . Let $D^{t\bullet }_{g,n}(\mu _1, \ldots , \mu _n) \in \mathbb {Q}[t]$ denote the analogous count, including possibly disconnected dessins d’enfant.

Example 4.6 Consider the dessin d’enfant enumeration $D^t_{0,2}(2,2)$ . The only dessins d’enfant that contribute are the five pictured below. We have represented these as plane graphs, with the face labeled 1 corresponding to the bounded region and the face labeled 2 corresponding to the unbounded region.

The central dessin d’enfant has two automorphisms, while the remaining dessins d’enfant have one, so we have $D^t_{0,2}(2,2) = \big ( \frac {t^3}{1} \big ) + \big ( \frac {t^2}{1} + \frac {t^2}{2} + \frac {t^2}{1} \big ) + \big ( \frac {t^1}{1} \big ) = t^3 + \frac {5}{2} t^2 + t$ .

Proposition 4.7 If $|\mu | < 2g + n$ , then $D_{g,n}^t(\mu _1, \ldots , \mu _n) = 0$ . If $|\mu | \geqslant 2g + n$ , then $D_{g,n}^t(\mu _1, \ldots , \mu _n) \in \mathbb {Q}[t]$ is a polynomial of degree $|\mu | + 1 - 2g - n$ whose coefficients are symmetric.

Proof Consider a dessin d’enfant with genus g and n faces with degrees $\mu _1, \ldots , \mu _n$ . By an Euler characteristic calculation, the number of vertices is $|\mu | + 2 - 2g - n$ . Since there must be at least one black vertex and at least one white vertex, we must have have $|\mu | \geqslant 2g + n$ .

We will show that, under the assumption $|\mu | \geqslant 2g + n$ , there exists a dessin d’enfant with exactly one white vertex. Take a polygon with $4g+2$ sides, whose vertices are alternately colored black and white. By identifying opposite edges, one obtains a dessin d’enfant on a genus g surface, with one black vertex, one white vertex, and one face with degree $2g+1$ . By adding $n-1$ appropriate edges from the black vertex to the white vertex, one can obtain a genus g dessin d’enfant with one black vertex, one white vertex, and n faces with any degrees $\mu _1, \ldots , \mu _n$ satisfying $|\mu | = 2g + n$ . We can relax this condition to $|\mu | \geqslant 2g + n$ by adding appropriate black vertices with degree 1 that are adjacent to the unique white vertex. Such a dessin d’enfant contributes to degree $|\mu | + 1 - 2g - n$ in $D_{g,n}^t(\mu _1, \ldots , \mu _n)$ , and there can be no contributions in higher degree.

Finally, observe that switching the colors of the vertices of a dessin d’enfant changes the number of black vertices from b to $(|\mu | + 2 - 2g - n) - b$ , leading to the desired symmetry in the coefficients of $D_{g,n}^t(\mu _1, \ldots , \mu _n)$ .

There is an analog of the cut-and-join recursion for deformed monotone Hurwitz numbers of Theorem 3.7 in the context of the dessin d’enfant enumeration. We omit the proof, since it can be found in the literature [Reference Kazarian and Zograf37].

Proposition 4.8 (Cut-and-join recursion)

Apart from the initial condition ${D^t_{0,1}(1) \,{=}\, 1}$ , the dessin d’enfant enumeration satisfies

$$ \begin{align*} &\mu_1 D^t_{g,n}(\mu_1, \mu_S) = \sum_{i=2}^n (\mu_1+\mu_i-1) \, D^t_{g,n-1}(\mu_1+\mu_i-1, \mu_{S \setminus \{i\}}) \\&\quad + (t+1) (\mu_1-1) \, D^t_{g,n}(\mu_1-1, \mu_S) \\&\quad + \sum_{\alpha + \beta = \mu_1-1} \alpha \beta \Bigg[ D^t_{g-1,n+1}(\alpha, \beta, \mu_S) + \sum_{\substack{g_1+g_2=g \\ I_1 \sqcup I_2 = S}} D^t_{g_1,|I_1|+1}(\alpha, \mu_{I_1}) \, D^t_{g_2,|I_2|+1}(\beta, \mu_{I_2}) \Bigg], \end{align*} $$

where we use the notation $S = \{2, 3, \ldots , n\}$ and $\mu _I = \{\mu _{i_1}, \mu _{i_2}, \ldots , \mu _{i_k}\}$ for $I = \{i_1, i_2, \ldots , i_k\}$ .

It is well-known that dessins d’enfant correspond to pairs of permutations acting on the edges—one permutation rotates edges around black vertices and the other rotates edges around white vertices. (Equivalent descriptions in the literature often use triples of permutations that compose to give the identity.) This leads to an expression for the disconnected enumeration of dessins d’enfant in terms of characters of the symmetric group, via Frobenius’s formula. For details, we recommend the book of Lando and Zonkin [Reference Lando and Zvonkin38]. To incorporate the parameter t, we observe that its exponent should equal the number of cycles in the permutation that rotates edges around black vertices. As a consequence of Jucys’s results described in part (a) of Proposition 2.13, we obtain the following result, for which we omit the proof.

Proposition 4.9 The disconnected enumeration of dessins d’enfant is given by the formula

$$\begin{align*}D_{g,n}^{t\bullet}(\mu_1, \ldots, \mu_n) = \frac{1}{\mu_1 \cdots \mu_n} \, \big[ \hbar^{2g-2+n} \big] \sum_{\lambda \in \mathcal{P}} \chi^\lambda_\mu \, s_\lambda(\tfrac{1}{\hbar}, \tfrac{1}{\hbar}, \ldots) \prod_{\square \in \lambda} \left( t + \hbar \, c(\square) \right). \end{align*}$$

We are now in a position to prove that topological recursion governs the weighted enumeration of dessins d’enfant.Footnote ⁵

Theorem 4.10 Topological recursion on the spectral curve

$$\begin{align*}\mathcal{C} = \mathbb{CP}^1, \quad x(z) = \frac{1 - z + tz}{z(1-z)}, \quad y(z) = z, \quad \omega_{0,2} = \frac{\mathrm{d} z_1 \, \mathrm{d} z_2}{(z_1-z_2)^2} \end{align*}$$

produces correlation differentials satisfying

$$ \begin{align*} \omega_{g,n}(z_1, \ldots, z_n) &= \mathrm{d}_1 \cdots \mathrm{d}_n \sum_{\mu_1, \ldots, \mu_n=1}^{\infty} D^t_{g,n}(\mu_1, \ldots, \mu_n) \, x(z_1)^{-\mu_1} \cdots x(z_n)^{-\mu_n} \\&\quad + \delta_{g,0} \delta_{n,2} \, \frac{\mathrm{d} x(z_1) \, \mathrm{d} x(z_2)}{\left( x(z_1) - x(z_2) \right)^2}. \end{align*} $$

Proof We again invoke the result of Bychkov, Dunin-Barkowski, Kazarian, and Shadrin [Reference Bychkov, Dunin-Barkowski, Kazarian and Shadrin8], noting that the more restricted result of Alexandrov, Chapuy, Eynard, and Harnad [Reference Alexandrov, Chapuy, Eynard and Harnad1] would suffice in this particular context. See the proof of Theorem 4.1 above for a statement of the result.

The representation-theoretic interpretation of the dessin d’enfant enumeration given in Proposition 4.9 implies that they are weighted Hurwitz numbers with the choice of data $\tilde {y}(z) = \sum _{i=1}^\infty z^i = \frac {z}{1-z}$ and $f(z) = t + z$ . Hence, topological recursion on the spectral curve

$$\begin{align*}\mathcal{C} = \mathbb{CP}^1, \quad x(z) = \frac{z(1-z)}{t-tz+z}, \quad y(z) = \frac{t-tz+z}{(1-z)^2}, \quad \omega_{0,2}(z_1, z_2) = \frac{\mathrm{d} z_1 \, \mathrm{d} z_2}{(z_1 - z_2)^2} \end{align*}$$

produces correlation differentials satisfying

$$ \begin{align*} \omega_{g,n}(z_1, \ldots, z_n) &= \mathrm{d}_1 \cdots \mathrm{d}_n \sum_{\mu_1, \ldots, \mu_n=1}^{\infty} D^t_{g,n}(\mu_1, \ldots, \mu_n) \, x(z_1)^{\mu_1} \cdots x(z_n)^{\mu_n} \\&\qquad + \delta_{g,0} \delta_{n,2} \, \frac{\mathrm{d} x(z_1) \, \mathrm{d} x(z_2)}{\left( x(z_1) - x(z_2) \right)^2}. \end{align*} $$

This does not yet match our spectral curve, since the enumeration is stored as an expansion at $x(z) = 0$ rather than at $x(z) = \infty $ . We obtain the desired spectral curve by performing the following three transformations in order:

$$\begin{align*}(x, y, t) \mapsto (x^{-1}, x^2y, t), \qquad (x, y, t) \mapsto (x, y+tx, t), \qquad (x, y, t) (tx, y, t^{-1}). \end{align*}$$

The first transformation changes the expansion at $x(z) = 0$ to an expansion at $x(z) = \infty $ ; the second transformation preserves the correlation differentials [Reference Eynard and Orantin28, Section 4.2]; and the third transformation preserves the coefficients of the expansion, due to the symmetry of the coefficients of $D^t_{g,n}(\mu _1, \ldots , \mu _n)$ and the homogeneity property of topological recursion [Reference Eynard and Orantin28, Section 4.1]. Thus, we arrive at the desired result.

Although the enumeration of dessins d’enfant has been studied previously [Reference Kazarian and Zograf37], the following conjectural properties have not yet been observed in the literature, to the best of our knowledge.

Conjecture 4.11 (Real-rootedness and interlacing)

For all $g \geqslant 0$ , $n \geqslant 1$ and $\mu _1, \ldots , \mu _n \geqslant 1$ , the dessin d’enfant enumeration $D_{g,n}^t(\mu _1, \mu _2, \ldots , \mu _n)$ is a real-rooted polynomial in t. Furthermore, the polynomial $D_{g,n}^t(\mu _1, \mu _2, \ldots , \mu _n)$ interlaces each of the n polynomials

$$\begin{align*}D_{g,n}^t(\mu_1+1, \mu_2, \ldots, \mu_n), \!\quad D_{g,n}^t(\mu_1, \mu_2+1, \ldots, \mu_n), \!\quad\! \ldots, \!\quad\! D_{g,n}^t(\mu_1, \mu_2, \ldots, \mu_n+1). \end{align*}$$

As with the deformed monotone Hurwitz numbers, we can effectively calculate $D_{g,n}^t(\mu _1, \ldots , \mu _n)$ using the cut-and-join recursion of Proposition 4.8 and explicitly check for real-rootedness and interlacing. Again, one finds a wealth of data to support Conjecture 4.11. We consider this as evidence that the real-rootedness and interlacing observed for deformed monotone Hurwitz numbers is not simply a sporadic artefact, but may be a more widespread phenomenon with a deep reason underlying it.

Remark 4.12. As previously mentioned, dessins d’enfant correspond to pairs of permutations acting on the edges, or equivalently, triples of permutations that compose to give the identity. It follows from Proposition 2.13 that each permutation has a unique strictly monotone factorization, defined analogously to a monotone factorization in Definition 2.10, but with the stronger monotonicity requirement that $b_1 < b_2 < \cdots < b_r$ . Using this result on the permutation that rotates edges around black vertices leads to the fact that the enumeration of dessins d’enfant can be interpreted as strictly monotone Hurwitz numbers.