A formula for enumerating permutations with a fixed pinnacle set

Diaz-Lopez, Alexander; Harris, Pamela E.; Huang, Isabella; Insko, Erik; Nilsen, Lars

doi:10.1016/j.disc.2021.112375

Cited by 11 publications

(5 citation statements)

References 17 publications

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“…Two articles investigating the number of interior peaks and the number of permutations with a given set of peak values, called "pinnacle set", are [42] and [18], respectively. To the best of our knowledge, the sum of the peaks and the sum of the squares of the peaks have not been yet investigated, despite of the recent interest in pinnacle sets [20,21,50]. In the next two sections, we will show that the moments of all statistics are also statistics, as defined above, and we will give closed forms for one of these moments for each of the statistics above.…”

Section: Definition 23mentioning

confidence: 92%

Moments of permutation statistics and central limit theorems

Dimitrov,

Khare

2021

Preprint

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We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns, which uses a lemma of Burstein and Hästö. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns. Closed forms for some of the higher moments of several popular statistics on permutations are also obtained.

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Section: Definition 23mentioning

confidence: 92%

Moments of permutation statistics and central limit theorems

Dimitrov,

Khare

2021

Preprint

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“…gave a recursive formula for the number of permutations in S n with a given pinnacle set p(n) and asked whether a more efficient expression could be computed. This paper led to a sequence of articles in recent years, many focused on improved and faster formulas for p(n), by realizing permutations with given pinnacle sets as invariants under certain modified S n -actions [5,9] or via more traditional enumerative methods [8,10,11]. In related work, Rusu [13] and Rusu-Tenner [14] deepened the knowledge of pinnacles in S n by investigating further properties of these statistics and characterizing admissible pinnacle orderings.…”

Section: Introductionmentioning

confidence: 99%

Pinnacle sets of signed permutations

González¹,

Harris²,

Kirby³

et al. 2023

Preprint

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Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for the type A symmetric group have been widely studied. In this article we define the pinnacle set of signed permutations of types B and D. We give a closed formula for the number of type B/D admissible pinnacle sets and answer several other related enumerative questions.

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“…Pinnacle sets of permutations have received great interest since their introduction in [3] by Davis, Nelson, Petersen, and Tenner. For the classical symmetric group, several properties of these sets have been studied [4,5,8,9,15,17,18]. More recently, the authors of this article characterized and enumerated pinnacle sets for signed permutations [11].…”

Section: Introductionmentioning

confidence: 99%