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Some generating functions and inequalities for the andrews–stanley partition functions
- Part of
-
- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 1 / February 2022
- Published online by Cambridge University Press:
- 27 December 2021, pp. 120-135
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- Article
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Let $\mathcal {O}(\pi )$
denote the number of odd parts in an integer partition $\pi$
. In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$
, where $\pi '$
is the conjugate of $\pi$
. Let $p(r,\,m;n)$
denote the number of partitions of $n$
with srank congruent to $r$
modulo $m$
. Generating function identities, congruences and inequalities for $p(0,\,4;n)$
and $p(2,\,4;n)$
were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for $p(r,\,m;n)$
with $m=16$
and $24$
. These results are refinements of some inequalities due to Swisher.
