The joint distribution of consecutive patterns and descents in permutations avoiding 3-1-2

Barnabei, Marilena; Bonetti, Flavio; Silimbani, Matteo

doi:10.1016/j.ejc.2009.11.011

Cited by 9 publications

(19 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A fuller investigation of the descent statistic over Sn(1-2-3), particularly its connection to Dyck paths, is given in [5].…”

Section: Methodsmentioning

confidence: 99%

Refining Enumeration Schemes to Count According to Permutation Statistics

Baxter

2014

Electron. J. Combin.

View full text Add to dashboard Cite

We consider the question of computing the distribution of a permutation statistics over restricted permutations via enumeration schemes. The restricted permutations are those avoiding sets of vincular patterns (which include both classical and consecutive patterns), and the statistics are described in the number of copies of certain vincular patterns such as the descent statistic and major index. An enumeration scheme is a polynomial-time algorithm (specifically, a system of recurrence relations) to compute the number of permutations avoiding a given set of vincular patterns. Enumeration schemes' most notable feature is that they may be discovered and proven via only finite computation. We prove that when a finite enumeration scheme exists to compute the number of permutations avoiding a given set of vincular patterns, the scheme can also compute the distribution of certain permutation statistics with very little extra computation.

show abstract

“…A fuller investigation of the descent statistic over Sn(1-2-3), particularly its connection to Dyck paths, is given in [5].…”

Section: Methodsmentioning

confidence: 99%

Refining Enumeration Schemes to Count According to Permutation Statistics

Baxter

2014

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…where now T ranges over all subsets of [n − 1] containing S and having no two consecutive elements. Equation (5) can also be obtained directly from Theorem 5.1 using Möbius inversion. For n, m ≥ 0, let…”

Section: Descent Sets On 321-avoiding Permutationsmentioning

confidence: 99%

“…In one of the first papers in this area, Robertson, Saracino and Zeilberger [24] considered the number of fixed points and excedances in permutations avoiding patterns of length 3, which sparked further work on these statistics by several authors [7,[16][17][18][19]. More recently, other statistics such as the number of descents [4,5], the major index and the number of inversions [10,15,26] have been studied on restricted permutations. Many of these papers show that certain statistics have the same distribution on permutations avoiding different patterns, and in some cases they give this distribution.…”

Section: Introductionmentioning

confidence: 99%

Descent sets on 321-avoiding involutions and hook decompositions of partitions

Barnabei

Bonetti

Elizalde

et al. 2014

Journal of Combinatorial Theory, Series A

Self Cite

View full text Add to dashboard Cite

We show that the distribution of the major index over the set of involutions in S n that avoid the pattern 321 is given by the q-analogue of the n-th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson-Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside a ⌊ n 2 ⌋ × ⌈ n 2 ⌉ box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions.

show abstract

“…There is also a number of papers studying other properties of random -avoiding permutations. Some examples, in addition to those mentioned above, are consecutive patterns [6]; descents and the major index [5]; number of fixed points [21,22,25,26,39,44]; position of fixed points [25,26,39]; exceedances [21,22]; longest increasing subsequence [19]; shape and distribution of individual values i [24,37,38].…”

Section: Introductionmentioning

confidence: 99%