A reciprocity method for computing generating functions over the set of permutations with no consecutive occurrence of a permutation pattern

Abstract

In this paper, we introduce a new method for computing generating functions with respect to the number of descents and left-to-right minima over the set of permutations which have no consecutive occurrence of τ where τ starts with 1. In particular, we study the generating function n≥0

Descent c-Wilf Equivalence

“…To prove Theorem 6, we will modify the reciprocity method of Jones and Remmel [15][16][17]. The reciprocity method was designed to study generating functions of the form…”

“…Jones and Remmel [17,18] showed that one can interpret n!θ τ (h n ) as a certain signed sum of the weights of filled, labeled brick tabloids when τ starts with 1 and des(τ ) = 1. They then defined a weightpreserving, sign-reversing involution I on the set of such filled, labeled brick tabloids which allowed them to give a relatively simple combinatorial interpretation for n!θ τ (n n ).…”

“…Then they showed how such a combinatorial interpretation allowed them to prove that for certain families of such permutations τ , the polynomials U τ,n (y) satisfy simple recursions. For example, in [17], Jones and Remmel studied the generating functions NM τ (t, x, y) for permutations τ of the form τ = 1324 • • • p where p ≥ 4. Using the reciprocity method, they proved that U 1324,1 (y) = −y and for n ≥ 2,…”

Generating functions for descents over permutations which avoid sets of consecutive patterns

“…Jones and Remmel [12][13][14] developed what they called the reciprocity method to compute the generating function NMτ (t, x, y) for certain families of permutations τ such that τ starts with 1 and des(τ ) = 1. The basic idea of their approach is as follows.…”

“…

We extend the reciprocity method of Jones and Remmel [13,14] to study generating functions of the form n≥0 t n n! σ∈N Mn(Γ) x LRmin(σ) y 1+des(σ) where Γ is a set of permutations which start with 1 and have at most one descent, N M n (Γ) is the set of permutations σ in the symmetric group S n which have no Γ-matches, des(σ) is the number of descents of σ and LRmin(σ) is the number of left-to-right minima of σ.

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