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Counting Pattern-Avoiding Permutations by Big Descents
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DOI：10.1007/s00026-026-00805-1.png)
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Abstract
En 中文
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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