Generating Functions for Permutations Avoiding a Consecutive Pattern

Abstract

Given a permutation τ of length j, we say that a permutation σ has a τ-match starting at position i, if the elements in positions i, i+1,... , i+ j −1 in σ have the same relative order as the elements of τ. We have been able to take advantage of the results of Mendes and Remmel [1] to obtain a generating function for the number of permutations with no τ-matches for several new classes of permutations. These new classes include a large class of permutations which are shuffles of an increasing sequence 1 2 ··· n with an arbitrary permutation σ of the integers {n + 1,..., n + m}. Finally we give a formula for the generating function for the number of permutations which have no 1 3 2 4-matches.

Clusters, generating functions and asymptotics for consecutive patterns in permutations

“…This pattern has been considered in [3,13]. In [3], Dotsenko and Khoroshkin give a recurrence for its cluster numbers r n,k .…”

“…This recurrence, which involves the Catalan numbers, is essentially equivalent to our derivation of equation ( 12) below. In [13], Liese and Remmel use a technique developed in [14] to obtain an ordinary generating function that is equivalent to R 1324 (−1, x). Here we find the differential equation satisfied by the bivariate generating function ω 1324 (u, z).…”

Computational Approaches to Consecutive Pattern Avoidance in Permutations

“…One of the early papers by Elizalde and Noy ( [5]) finds generating functions A(z) and P (z, t) for certain cases of single pattern avoidance. Using various techniques, additional generating functions for specific single patterns and multi-pattern sets have been found in [1,3,8,9]. In particular, our approach will resemble the cluster method approach in [3].…”

A survey of consecutive patterns in permutations

“…This power series is reminiscent of the cluster generating function R σ (−1, z), and in the case of non-overlapping patterns it actually coincides with it [25]. Subsequently, Liese and Remmel [54] gave explicit expressions for this power series in some cases where σ is a shuffle of an increasing sequence with another pattern.…”