Generating functions for permutations which avoid consecutive patterns with multiple descents
arXiv: Combinatorics(2017)
摘要
Let S_n denote the group all permutations of n. For every permutation σ, we let des(σ) denote the number of descents in σ and LRMin(σ) denote the number of left-to-right minima of σ. Given a sequence τ = τ_1 ⋯τ_n of distinct positive integers, we define the reduction of τ, red(τ), to be the permutation of S_n that results by replacing the i-th smallest element of τ by i. If Γ is a set of permutations, we say that a permutation σ = σ_1 …σ_n ∈ S_n has a Γ-match starting at position i if there is a i < j such that red(σ_i σ_i+1…σ_j) ∈Γ. We let Γ-mch(σ) denote the number of Γ-matches in σ. We let 𝒩ℳ_n(Γ) be the set of σ∈ S_n such that Γ-mch(σ) = 0. In this paper, we modify Jones and Remmel's reciprocity method to study the generating function of the form _Γ(t,x,y)=∑_n ≥ 0t^n/n!_Γ,n(x,y) where _Γ,n(x,y) =∑_σ∈𝒩ℳ_n(Γ)x^LRmin(σ)y^1+des(σ) in the case where we no longer insist that all the permutations τ∈Γ have at most one descent.
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