On Simsun and Double Simsun Permutations Avoiding a Pattern of Length Three

Abstract

A permutation σ ∈ Sn is simsun if for all k, the subword of σ restricted to {1, . . . , k} does not have three consecutive decreasing elements. The permutation σ is double simsun if both σ and σ −1 are simsun. In this paper we present a new bijection between simsun permutations and increasing 1-2 trees, and show a number of interesting consequences of this bijection in the enumeration of pattern-avoiding simsun and double simsun permutations. We also enumerate the double simsun permutations that avoid each pattern of length three.

More bijections for Entringer and Arnold families

“…In Section 5, we shall give more insightful description of two important bijections. More precisely, Chuang et al's constructed a φ : T n+1 → RS n in [4], we show that φ can be factorized as the compositions of two of our simpler bijections, and then give a direct description of Gelineau et al's bijection ψ : DU n → T n in [11].…”

“…Let DU n be the set of (down-up) alternating permutations of [n]. For example, n \ k 1 2 3 4 In 1933 Kempener [14] used the boustrophedon algorithm (2) to enumerate alternating permutations without refering to Euler numbers. Since Entringer [7] first found the combinatorial interpretation of Kempener's table (E n,k ) in terms of André's model for Euler numbers, the numbers E n,k are then called Entringer numbers.…”

“…Arnold) families with the known ones. We refer the reader to the more recent papers [23,11,4,13,8] related to the combinatorics of Euler numbers and Springer numbers. This paper is organized as follows.…”

“…For example, the permutation σ = 43512 is not André since the subword σ [4] = 4312 contains a double descent (4, 3, 1), while the permutation τ = 31245 is André since there is no double descent in the subwords:…”

The Peak Statistics on Simsun Permutations

“…In this paper we study the peak statistics on simsum permutations. It is well known that the descent statistic is equidistributed over n-simsun permutations and n-André permutations (see [5]), and there are bijections between simsun permutations and increasing 1-2 trees (see [6] for instance). Therefore, one can find corresponding results on André permutations and increasing 1-2 trees.…”