Dyck paths and restricted permutations

Inversion Formulae on Permutations Avoiding 321

“…where C n is the n-th Catalan number which counts the number of Dyck paths of length 2n. In past decades, various articles considered the bijections between 321-avoiding permutations and Dyck paths, see [4,7,10,11,14,15,17,18,19,21,22]. In this paper, we will study the inversion distribution of 321-avoiding permutations.…”

“…Soon afterwards a bijective proof of the recursive formula (1) was obtained by Szu-En Cheng et al [6]. There are some other works on inversions of restricted permutations, see [1,3,5,9,15,16]. In 2014, M. Barnabei, F. Bonetti, S. Elizalde and M. Silimbani [2] studied the distribution of descents and major indexes of 321-avoiding involutions.…”

Reduced Decompositions of Matchings

“…The systematic study of pattern avoiding permutations was initiated in 1985 [17]. Starting with the work of Billey, Jockusch and Stanley [3], there has been increasing interest in the connection between reduced decomposition and pattern avoiding permutation (see [1,2,16,19] and references therein). Other results involving pattern avoiding matchings appeared in [5-7, 9-15, 20, 21].…”

Partial matchings and pattern avoidance

“…f (t) = 1 + t + 2t 2 + 4t 3 + 10t 4 + 25t 5 + 67t 6 + 182t 7 + 512t 8 + 1460t 9 + 4241t 10 + 12453t 11 + 36999t 12 + 110865t 13 + 334929t 14 + 1018545t 15 …”

“…Suppose first that m occurs once in σ ′ . There are then m − 1 n − m choices for the letters of [m − 1] appearing in σ ′ and c n−m+1 ways in which to order the letters of σ ′ since they must form a 123-avoiding permutation of length n − m + 1 (see, e.g.,[13] or[15]). Thus, there arem − 1 n − m c n−m+1 members of V n,m (1123) in this case.…”