vol. 27:1, Permutation Patterns 2024

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1. Sorting inversion sequences

Toufik Mansour ; Howard Skogman ; Rebecca Smith.

We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and generalizations, some of which we apply here. We give the classification of several classes of sortable inversion sequences in terms of pattern avoidance. We are able to provide an exact enumeration of some of the sortable classes in question using both classical approaches and a more recent strategy utilizing generating trees.

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2. Key-avoidance for alternating sign matrices

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.

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