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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