Refined Restricted Permutations

Abstract

Dedication: In memory of Rodica Simion This article is dedicated to the memory of Rodica Simion, one of the greatest enumerators of the 20 th century. Both derangements [8] and restricted permutations [10] were very dear to her heart, and we are sure that she would have appreciated the present surprising connections between these at-first-sight unrelated concepts.Abstract. Define S k n (α) to be the set of permutations of {1, 2,...,n} with exactly k fixed points which avoid the pattern α ∈ S m . Let s k n (α) be the size of S k n (α). We investigate S 0 n (α) for all α ∈ S 3 as well as show that s k n (132) = s k n (213) = s k n (321) and s k n (231) = s k n (312) for all 0 ≤ k ≤ n.

Bijections for refined restricted permutations

“…In this paper we present a bijective proof of the ''refined'' results on 321-and 132-avoiding permutations, resolving the problem which was left open in [10,4]. In fact, our bijection is a composition of two (slightly modified) known bijections into Dyck paths, and the result follows from a new analysis of these bijections.…”

“…An unexpected recent result of Robertson et al [10] gives a new and exciting extension to what is now regarded as a classical result that the number of 321-avoiding permutations equals the number of 132-avoiding permutations. They show that one can ''refine'' this result by taking into account the number of fixed points in a permutation.…”

“…Theorem 1 (Robertson et al [10], Elizalde [4]). The number of 321-avoiding permutations sAS n with fpðsÞ ¼ i and excðsÞ ¼ j equals the number of 132-avoiding permutations sAS n with fpðsÞ ¼ i and excðsÞ ¼ j; for any 0pi; jpn:…”

Two refined major-balance identities on 321-avoiding involutions

“…Simion and Gessel studied the refined sign-balance on S n , respecting the major index (see [12,Corollary 2]). Robertson, Saracino and Zeilberger [10] started the refined enumeration of pattern-avoiding permutations, respecting the number of fixed points and excedances. These results sparked further work on refined signed enumeration for restricted permutations.…”

A Simple and Unusual Bijection for Dyck Paths and its Consequences

“…Theorem 6.1. [7] The number of 321-avoiding permutations π ∈ S n with fp(π) = k equals the number of 132-avoiding permutations π ∈ S n with fp(π) = k, for any 0 ≤ k ≤ n.…”