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The dimension of the feasible region of pattern densities

Part of: Extremal combinatorics Classical measure theory Combinatorics

Published online by Cambridge University Press:  09 January 2025

FREDERIK GARBE ,
DANIEL KRÁL’ ,
ALEXANDRU MALEKSHAHIAN  and
RAUL PENAGUIAO
FREDERIK GARBE
Affiliation:
Institute of Computer Science, Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany. e-mail: garbe@informatik.uni-heidelberg.de
DANIEL KRÁL’
Affiliation:
Faculty of Informatics, Masaryk University, Botanická 68A, 602 00 Brno, Czech Republic. e-mail: dkral@fi.muni.cz
ALEXANDRU MALEKSHAHIAN
Affiliation:
Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK. e-mail: alexandru.malekshahian@kcl.ac.uk
RAUL PENAGUIAO
Affiliation:
Max Planck Institute for the Sciences, Inselstraße 22, 04103 Leipzig, Germany. e-mail: raul.penaguiao@mis.mpg.de
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Abstract

A classical result of Erdős, Lovász and Spencer from the late 1970s asserts that the dimension of the feasible region of densities of graphs with at most k vertices in large graphs is equal to the number of non-trivial connected graphs with at most k vertices. Indecomposable permutations play the role of connected graphs in the realm of permutations, and Glebov et al. showed that pattern densities of indecomposable permutations are independent, i.e., the dimension of the feasible region of densities of permutation patterns of size at most k is at least the number of non-trivial indecomposable permutations of size at most k. However, this lower bound is not tight already for $k=3$. We prove that the dimension of the feasible region of densities of permutation patterns of size at most k is equal to the number of non-trivial Lyndon permutations of size at most k. The proof exploits an interplay between algebra and combinatorics inherent to the study of Lyndon words.

MSC classification

Primary: 05A05: Permutations, words, matrices
Secondary: 05D99: None of the above, but in this section 28A12: Contents, measures, outer measures, capacities
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 14
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

An extended abstract announcing the results presented in this paper has been published in the Proceedings of Eurocomb’23.

‡

Previous affiliation: Faculty of Informatics, Masaryk University, Botanická 68A, 602 00 Brno, Czech Republic. Supported by the MUNI Award in Science and Humanities (MUNI/I/1677/2018) of the Grant Agency of Masaryk University.

§

Supported by the MUNI Award in Science and Humanities (MUNI/I/1677/2018) of the Grant Agency of Masaryk University.

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