Frame patterns in n-cycles

Abstract

Abstract. In this paper, we study the distribution of the number occurrences of the simplest frame pattern, called the µ pattern, in n-cycles. Given an n-cycle C, we say that a pair i, j matches the µ pattern if i < j and as we traverse around C in a clockwise direction starting at i and ending at j, we never encounter a k with i < k < j. We say that i, j is a nontrivial µ-match if i + 1 < j. We say that an n-cycle C is incontractible if there is no i such that i + 1 immediately follows i in C. We show that number of incontractible n-cycles in the symmetric group Sn is Dn−1 where Dn is the number of derangements in Sn. We show that number of n-cycles in Sn with exactly k µ-matches can be expressed as a linear combination of binomial coefficients of the form n−1 i where i ≤ 2k + 1. We also show that the generating function N T In,µ(q) of q raised to the number of nontrivial µ-matches in C over all incontractible n-cycles in Sn is a new qanalogue of Dn−1 which is different from the q-analogues of the derangement numbers that have been studied by Garsia and Remmel and by Wachs. We will show that there is a rather surprising connection between the charge statistic on permutations due to Lascoux and Schüzenberger and our polynomials in that the coefficient of the smallest power of q in N T I 2k+1,µ (q) is the number of permutations in S 2k+1 whose charge path is a Dyck path. Finally, we show that N T In,µ(q)| q ( n−12 )−k and N Tn,µ(q)| q ( n−12 )−k are the number of partitions of k for sufficiently large n.

Distributions of several infinite families of mesh patterns

“…The notion of a mesh pattern, generalizing several classes of patterns, was introduced by Brändén and Claesson [3] to provide explicit expansions for certain permutation statistics as, possibly infinite, linear combinations of (classical) permutation patterns. Several papers are dedicated to the study of mesh patterns and their generalizations [1,2,5,7,8,9,13,14].…”

Asymptotic Behaviour of the Containment of Certain Mesh Patterns

“…Of particular interest in permutation patterns is the enumeration of the avoidance or containment class of particular permutations. Mesh patterns have been studied extensively since their introduction, see e.g., [AKV13,Ten13,CTU15,JKR15,BGMU19]. The first systematic study of the avoidance of mesh patterns was conducted in [HJS + 15], where enumeration results were given for the avoidance of 25 patterns of length 2.…”

The poset of mesh patterns

“…Mesh patterns were first introduced by Brändén and Claesson in [BC11] as a generalisation of permutation patterns, and have been studied extensively in recent years, see e.g., [CTU15,JKR15]. A mesh pattern consist of a pair (π, P ), where π is a permutation and P is a set of coordinates in a square grid.…”