Mathilde Bouvel ; Rebecca Smith ; Jessica Striker - Key-avoidance for alternating sign matrices

dmtcs:14058 - Discrete Mathematics & Theoretical Computer Science, March 13, 2025, vol. 27:1, Permutation Patterns 2024 - https://doi.org/10.46298/dmtcs.14058
Key-avoidance for alternating sign matricesArticle

Authors: Mathilde Bouvel ; Rebecca Smith ; Jessica Striker

    We initiate a systematic study of key-avoidance on alternating sign matrices (ASMs) defined via pattern-avoidance on an associated permutation called the \emph{key} of an ASM. We enumerate alternating sign matrices whose key avoids a given set of permutation patterns in several instances. We show that ASMs whose key avoids 231 are permutations, thus any known enumeration for a set of permutation patterns including 231 extends to ASMs. We furthermore enumerate by the Catalan numbers ASMs whose key avoids both 312 and 321. We also show ASMs whose key avoids 312 are in bijection with the gapless monotone triangles of [Ayyer, Cori, Gouyou-Beauchamps 2011]. Thus key-avoidance generalizes the notion of 312-avoidance studied there. Finally, we enumerate ASMs with a given key avoiding 312 and 321 using a connection to Schubert polynomials, thereby deriving an interesting Catalan identity.


    Volume: vol. 27:1, Permutation Patterns 2024
    Section: Special issues
    Published on: March 13, 2025
    Accepted on: February 11, 2025
    Submitted on: August 13, 2024
    Keywords: Mathematics - Combinatorics,05A05, 05A19

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