The largest singletons in weighted set partitions and its applications

TL;DR

This paper explores the weighted set partitions focusing on the largest singleton elements, deriving explicit formulas and identities, and applies these findings to permutations, involutions, and forests, including a novel identity related to tree enumeration.

Contribution

It introduces explicit formulas and combinatorial identities for weighted set partitions, extending the analysis to permutations, involutions, and forests, with a new identity analogous to Riordan's.

Findings

Derived explicit formulas for weighted set partitions with largest singleton

Established combinatorial identities involving these formulas

Discovered a new identity related to tree enumeration and derangements

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 weighted set partitions. Let $A_{n,k}(\mathbf{t})$ denote the total weight of partitions on $[n+1]$ with the largest singleton $\{k+1\}$. In this paper, explicit formulas for $A_{n,k}(\mathbf{t})$ and many combinatorial identities involving $A_{n,k}(\mathbf{t})$ are obtained by umbral operators and combinatorial methods. As applications, we investigate three special cases such as permutations, involutions and labeled forests. Particularly in the permutation case, we derive a surprising identity analogous to the Riordan identity related to tree enumerations, namely, \begin{eqnarray*} \sum_{k=0}^{n}\binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, \end{eqnarray*} where $D_{k}$ is the $k$-th derangement number or the number of permutations of $\{1,2,\dots, k\}$ with no fixed points.