Counting pattern-avoiding permutations by big descents
S Elizalde, J Rivera Jr, Y Zhuang - arXiv preprint arXiv:2408.15111, 2024 - arxiv.org
S Elizalde, J Rivera Jr, Y Zhuang
arXiv preprint arXiv:2408.15111, 2024•arxiv.orgA descent $ k $ of a permutation $\pi=\pi_ {1}\pi_ {2}\dots\pi_ {n} $ is called a\textit {big
descent} if $\pi_ {k}>\pi_ {k+ 1}+ 1$; denote the number of big descents of $\pi $ by
$\operatorname {bdes}(\pi) $. We study the distribution of the $\operatorname {bdes} $
statistic over permutations avoiding prescribed sets of length-three patterns. Specifically, we
classify all pattern sets $\Pi\subseteq\mathfrak {S} _ {3} $ of size 1 and 2 into
$\operatorname {bdes} $-Wilf equivalence classes, and we derive a formula for the …
descent} if $\pi_ {k}>\pi_ {k+ 1}+ 1$; denote the number of big descents of $\pi $ by
$\operatorname {bdes}(\pi) $. We study the distribution of the $\operatorname {bdes} $
statistic over permutations avoiding prescribed sets of length-three patterns. Specifically, we
classify all pattern sets $\Pi\subseteq\mathfrak {S} _ {3} $ of size 1 and 2 into
$\operatorname {bdes} $-Wilf equivalence classes, and we derive a formula for the …
A descent of a permutation is called a \textit{big descent} if ; denote the number of big descents of by . We study the distribution of the statistic over permutations avoiding prescribed sets of length-three patterns. Specifically, we classify all pattern sets of size 1 and 2 into -Wilf equivalence classes, and we derive a formula for the distribution of big descents for each of these classes. Our methods include generating function techniques along with various bijections involving objects such as Dyck paths and binary words. Several future directions of research are proposed, including conjectures concerning real-rootedness, log-concavity, and Schur positivity.
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