Abstract A descent k of a permutation = ₁ ₂ ₍ π = π 1 π 2 ⋯ π n is called a big descent if ₊> ₊+₁+1 π k > π k + 1 + 1 ; denote the number of big descents of π by {\, bdes\, } () bdes (π). We study the distribution of the {\, bdes\, } bdes statistic over permutations avoiding prescribed sets of length-three patterns. Specifically, we classify all pattern sets S₃ Π ⊆ S 3 of size 1 and 2 into {\, bdes\, } 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 directions of research are proposed, including conjectures concerning real-rootedness, log-concavity, and Schur positivity.
Elizalde et al. (Mon,) studied this question.
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