Abstract
We find an explicit expression for the generating function of the number of permutations in S_n avoiding a subgroup of S_k generated by all but one simple transpositions. The generating function turns out to be rational, and its denominator is a rook polynomial for a rectangular board.
Highlights
Introduction and Main ResultLet [p] = {1, . . . , p} denote a totally ordered alphabet on p letters, and let α = (α1, . . . , αm) ∈ [p1]m, β = (β1, . . . , βm) ∈ [p2]m
A natural generalization of single pattern avoidance is subset avoidance; that is, we say that π ∈ Sn avoids a subset T ⊂ Sk if π avoids any τ ∈ T
We denote by Pl,m the parabolic subgroup of Sl+m generated by s1, . . . , sl−1, sl+1, . . . , sl+m−1, and by fl,m(n) the number of permutations in Sn avoiding all the patterns in Pl,m
Summary
We find an explicit expression for the generating function of the number of permutations in Sn avoiding a subgroup of Sk generated by all but one simple transpositions.
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