Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Permutation Statistics and Pattern Avoidance in Involutions   Dokos et. al. studied the distribution of two statistics over permutations
$\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three
patterns. A permutation $\sigma\in\mathfrak{S}_n$ contains a pattern
$\pi\in\mathfrak{S}_k$ if $\sigma$ has a subsequence of length $k$ whose
letters are in the same relative order as $\pi$. This paper is a comprehensive
study of the same two statistics, number of inversions and major index, over
involutions $\mathcal{I}_n=\{\sigma\in\mathfrak{S}_n:\sigma^2=\text{id}\}$ that
avoid one or more length three patterns. The equalities between the generating
functions are consequently determined via symmetries and we conjecture this
happens for longer patterns as well. We describe the generating functions for
each set of patterns including the fixed-point-free case, $\sigma(i)\neq i$ for
all $i.$ Notating $M\mathcal{I}_n(\pi)$ as the generating function for the
major index over the avoidance class of involutions associated to $\pi$ we
particularly present an independent determination that $M\mathcal{I}_n(321)$ is
the $q$-analogue for the central binomial coefficient that first appeared in a
paper by Barnebei, Bonetti, Elizalde and Silimbani. A shorter proof is
presented that establishes a connection to core, a central topic in poset
theory. We also prove that
$M\mathcal{I}_n(132;q)=q^{\binom{n}{2}}M\mathcal{I}_n(213;q^{-1})$ and that the
same symmetry holds for the larger class of permutations conjecturing that the
same equality is true for involutions and permutations given any pair of
patterns of the form $k(k-1)\dots 1(k+1)(k+2)\dots m$ and $12\dots (k-1)
m(m-1)\dots k$, $k\leq m$.

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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat