Permutations Avoiding a Simsun Pattern

Marilena Barnabei,Matteo Silimbani,Flavio Bonetti,Niccolò Castronuovo

doi:10.37236/9482

Abstract

A permutation $\pi$ avoids the simsun pattern $\tau$ if $\pi$ avoids the consecutive pattern $\tau$ and the same condition applies to the restriction of $\pi$ to any interval $[k].$ Permutations avoiding the simsun pattern $321$ are the usual simsun permutation introduced by Simion and Sundaram. Deutsch and Elizalde enumerated the set of simsun permutations that avoid in addition any set of patterns of length $3$ in the classical sense. In this paper we enumerate the set of permutations avoiding any other simsun pattern of length $3$ together with any set of classical patterns of length $3.$ The main tool in the proofs is a massive use of a bijection between permutations and increasing binary trees.

Highlights

A permutation σ ∈ Sn avoids the pattern τ ∈ Sk if there are no indices i1, i2, . . . , ik such that the subsequence σi1σi2 . . . σik is order isomorphic to τ.A permutation σ is called simsun if it does not contain double descents and the same applies to the restriction of σ to any interval [k]
The theory of simsun permutation goes back to the work by Sundaram [13] where the author proved that the cardinality of the set of simsun permutations of length n is the (n + 1)−th Euler number
We deal with a generalization of simsun permutations defined by Lin, Ma, and Yeh ([7]): we say that a permutation σ avoids the simsun pattern τ if the restriction of σ to the interval [k] does not contain the consecutive pattern τ for any k = 1, . . . , n

Summary

Introduction

A permutation σ ∈ Sn avoids the (classical) pattern τ ∈ Sk if there are no indices i1, i2, . . . , ik such that the subsequence σi1σi2 . . . σik is order isomorphic to τ. The theory of simsun permutation goes back to the work by Sundaram [13] where the author proved that the cardinality of the set of simsun permutations of length n is the (n + 1)−th Euler number (see sequence A000111 in [11]). We denote by Sn(τ S) the set of all permutations in Sn that avoid the simsun pattern τ. For all Σ, we can partition the set of simsun patterns of length 3 into three classes with respect to the avoidance of the classical patterns in Σ:. We observe that a permutation π avoids the simsun pattern 132 if and only if each occurrence of 132 in π is part of an occurrence of 2413 or, equivalently, π avoids the barred pattern 2413 It has been shown in [2] that these permutations are enumerated by the Bell numbers.

Permutations avoiding the simsun pattern 213

Connection with barred generalized patterns