Counting Pattern-Avoiding Permutations by Big Descents

doi:10.1007/s00026-026-00805-1

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Counting Pattern-Avoiding Permutations by Big Descents

2026-02-01

Elizalde, Sergi

Rivera, Johnny

Zhuang, Yan *

DOI：10.1007/s00026-026-00805-1

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Abstract

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A descent k of a permutation pi = pi(1)pi(2) pi(n) is called a big descent if pi(k) > pi(k +1) +1; denote the number of big descents of pi by bdes(pi). We study the distribution of the bdes statistic over permutations avoiding prescribed sets of length-three patterns. Specifically, we classify all pattern sets Pi subset of S-3 of size 1 and 2 into bdes-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 d pi rections of research are proposed, including conjectures concerning real-rootedness, log-concavity, and Schur positivity.

Keywords:

Permutation patterns

Big descents

st-Wilf equivalence

Dyck paths

Binary words

Narayana numbers

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