The largest singletons of set partitions

TL;DR

This paper explores the enumeration and properties of set partitions with a focus on the largest singleton element, providing explicit formulas, combinatorial identities, and investigating periodicity and congruence properties.

Contribution

It introduces new explicit formulas and identities for the counts of set partitions with a specified largest singleton, extending the understanding of their algebraic and combinatorial structure.

Findings

Derived explicit formulas for $A_{n,k}$ involving Dobinski-type analogs.

Established combinatorial identities and explored congruence properties.

Proved sequences are periodic modulo primes and conjectured their minimal periods.

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 $\{1,2,\dots, n+1\}$ with the largest singleton $\{k+1\}$ for $0\leq k\leq n$. 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 $(A_{n+k,k})_{n\geq 0}$ and $(A_{n+k,k})_{k\geq 0}$ (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 $N_p=\frac{p^p-1}{p-1}$ for any prime $p$.