The largest singletons of set partitions

Abstract

Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in set partitions. Let A n,k denote the number of partitions of {1, 2, . . . , n + 1} with the largest singleton {k + 1} for 0 ≤ k ≤ n. In this paper, several explicit formulas for A n,k , involving a Dobinski-type analog, are obtained by algebraic and combinatorial methods, many combinatorial identities involving A n,k and Bell numbers are presented by operator methods, and congruence properties of A n,k are also investigated. It will been showed that the sequences (A n+k,k ) n≥0 and (A n+k,k ) k≥0 (mod p) are periodic for any prime p, and contain a string of p − 1 consecutive zeroes. Moreover their minimum periods are conjectured to be N p = p p −1 p−1 for any prime p.

Congruences on the Bell polynomials and the derangement polynomials

“…where V n = n k=0 (−1) n−k n k B k is the number of partitions of [n] without singletons (i.e., one-element subsets) [9].…”

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

“…Hence, the two expressions for υ 2k equal. The final equality in parts (b) and (d) is a well-known property of Bell numbers (see for example [SW,(1.2…”

“…3.5], υ ℓ + υ ℓ+1 = B(ℓ), (the ℓth Bell number). Now [SW,Sec. 1] implies that υ 2k = 2k ℓ=0 (−1) 2k−ℓ 2k ℓ B(ℓ).…”

Asymptotic approximations of the continuous Hahn polynomials and their zeros

“…The Touchard polynomials Q n (x) and Pasternack-Bateman polynomials F λ n (x) are used in the Hermite-Padé approximations of the exponential functions and in Prévost's proof of the irrationality of the ζ(2) and ζ(3), where ζ(x) is the Riemann zeta function; see [29,30]. A connection between the polynomials of Q n (x) and the combinatorial problem of set partitions can be found in [31,33].…”