Permutations and words counted by consecutive patterns

Abstract

Generating functions which count occurrences of consecutive sequences in a permutation or a word which matches a given pattern are studied by exploiting the combinatorics associated with symmetric functions. Our theorems take the generating function for the number of permutations which do not contain a certain pattern and give generating functions refining permutations by the both the total number of pattern matches and the number of non-overlapping pattern matches. Our methods allow us to give new proofs of several previously recorded results on this topic as well as to prove new extensions and new q-analogues of such results.

Generating Functions for Permutations Avoiding a Consecutive Pattern

“…Remmel and Mendes [1] proved Theorem 1.1 by applying a ring homomorphism defined on the ring Λ of symmetric functions over infinitely many variables x 1 , x 2 ,... to a simple symmetric function identity. Kitaev's original proof of Theorem 1.1 was completely different and was actually more simple than the Mendes-Remmel proof.…”

“…The Mendes-Remmel proof has the advantage that it allowed them to give several extensions of Theorem 1.1 and the formula for A τ (t) referred to in (1.6) was produced as a byproduct of the proof. We will not present the Mendes-Remmel proof here as it can be found in [1]. The main goal of this section is to review the definitions in [1] that are needed to define all the terms that appear in (1.6).…”

“…We will not present the Mendes-Remmel proof here as it can be found in [1]. The main goal of this section is to review the definitions in [1] that are needed to define all the terms that appear in (1.6).…”

“…That is, instead of counting the number of permutations in S n with τ-mch σ = 0, we may try to understand the sum on the right hand side of the statement of the corollary. The following example was provided in [1] which we shall review as some of our new results generalize this example.…”

“…We shall also assume that red β (1) , red β (2) ,... , red β (k) are pairwise distinct permutations. We first consider the permutations τ = τ r 1 , β (1) ,..., r k , β (k) , r k+1 where r 1 ≥ r k+1 .…”