Set partition asymptotics and a conjecture of Gould and Quaintance

Abstract

The main result of this paper is the generalization and proof of a conjecture by Gould and Quaintance on the asymptotic behavior of certain sequences related to the Bell numbers. Thereafter we show some applications of the main theorem to statistics of partitions of a finite set S, i.e., collections B 1 , B 2 , . . . , B k of non-empty disjoint subsets of S such that k i=1 B i = S, as well as to certain classes of partitions of [n].

Counting water cells in bargraphs of compositions and set partitions

“…We treat the case when k is even, the odd case being similar which we leave to the reader. Let P * n,k denote the subset of P n,k consisting of π = 1π (1) 2π (2) • • • kπ (k) satisfying the following conditions: (i) π (2) = π (4)…”

“…By [2], the coefficient of x n n! in e e x x 0 e −e t dt is given by B n (C +O(e −κn/ log n )), where C = 0.5963473622 • • • and κ is a positive constant.…”

“…Given 1 ≤ i ≤ k ≤ n, let A n,k,i denote the set of non-nesting partitions of [n] having k blocks ending in the letter i and let A n,k = ∪ k i=1 A n,k,i . We represent π ∈ A n,k as π = 1π (1) 2π (2) • • • kπ (k) , where each π ( ) is -ary. Let a(n, k, i) = a p,q (n, k, i) denote the joint distribution on A n,k,i for the statistics recording the length of π (k) and the wc-value (marked by p and q, respectively).…”

Counting subwords in flattened partitions of sets

“…Once again, we make use of the saddle point method and the quasi-power theorem (Theorems 2.9 and 2.11). One can argue as in [1] to show that the integral in the generating function (12) can be extended to the range (0, ∞) at the expense of a small error term in the coefficients (cf. the proof of Corollary 2.24).…”

“…We can focus on the part −e r −r dr dt of the generating function, since we have exact formulas for the rest. Integration by parts yields contributes O(1). For the rest, we can use a general result recently proven in[1]: namely, if g is an entire function in the complex plane with g(z) = O(e e (1−ϵ)|z| ) for some ϵ > 0 as |z| → ∞, then the coefficients of…”