Asymptotics of the extremal excedance set statistic

Abstract

Abstract. For non-negative integers r and s with r + s = n − 1, let [b r a s ] denote the number of permutations π ∈ Sn which have the property that π(i) > i if, and only if, i ∈ [r] = {1, . . . , r}. Answering a question of Clark and Ehrenborg (2010), we determine asymptotics on [b r a s ] when r = (n − 1)/2 :We also determine asymptotics on [b r a s ] for a suitably related r = Θ(s) → ∞. Our proof depends on multivariate asymptotic methods of R. Pemantle and M. C. Wilson.We also consider two applications of our main result. One, we determine asymptotics on the number of permutations π ∈ Sn which simultaneously avoid the vincular patterns 21-34 and 34-21, i.e., for which π is order-isomorphic to neither (2,1,3,4) nor (3, 4, 2, 1) on any coordinates 1 ≤ i < i+1 < j < j +1 ≤ n. We also determine asymptotics on the number of n-cycles π ∈ Cn which avoid stretching pairs, i.e., those for which 1 ≤ π(i) < i < j < π(j) ≤ n.

Where can a place cell put its fields? Let us count the ways

“…Poly-Bernoulli numbers were originally introduced by Kaneko to enumerate the set of binary k-by-n matrices that are uniquely reconstructible from their row and column sums Kaneko (1997). The use of Poly-Bernoulli numbers in enumerating permutations of Young diagrams was pioneered in Postnikov (2006), while their asymptotic behavior along the diagonal was obtained in de Andrade et al (2015) as:…”

Combinatorics of poly-Bernoulli numbers

“…The diagonal sum of poly-Bernoulli numbers resp. their relatives arise in analytical, number theoretical and combinatorial investigations [33], [26], [2]. However a nice formula is still missing.…”

“…Cycles without stretching pairs [2] received attention because of their connection to a result of Sharkovsky in discrete dynamical systems. The occurrence of a stretching pair within a periodic orbit implies turbulence [17].…”

Asymptotic enumeration of lonesum matrices

“…Asymptotics for the poly-Bernoulli relative C n,k follow from the bijections in [3] along with the asymptotics of certain permutation statistics given in [5], where the authors applied the general machinery of analytic combinatorics in several variables (ACSV) developed by Pemantle, Wilson, and Baryshnikov [12].…”

“…We first use the more classical techniques from [10] to prove Theorem 2.1 in Section 2; this method only applies to the diagonal case n = k corresponding to square lonesum matrices. We then adapt the method from [5] in order to establish a more general bivariate asymptotic when n, k → ∞ with n/k varying within an arbitrary compact subset of the positive real numbers, see Theorem 3.1 in Section 3.…”