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We study the restricted growth function associated with set partitions, and obtain exact formulas for the number of strong records with height one, the total of record heights over set of partitions, and the number of partitions with a given maximal height of strong records. We also extend some of these results to weak records.
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 ${1,2,dots, n+1}$ with t
We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of stein (Arxiv:math.CO/0605670).
Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let $A_{n,k}(mathbf{t})$ denote the total weight of partitions o
The symmetric group $mathfrak{S}_n$ acts on the polynomial ring $mathbb{Q}[mathbf{x}_n] = mathbb{Q}[x_1, dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $mathfrak{S}_n$-invariant polynomials with vanishing
A seminal theorem of Tverberg states that any set of $T(r,d)=(r-1)(d+1)+1$ points in $mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any collection of fewer points in $mathbb{R}^d$ can