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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.
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
Inspired by Andrews 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations of
We prove three main conjectures of Berkovich and Uncu (Ann. Comb. 23 (2019) 263--284) on the inequalities between the numbers of partitions of $n$ with bounded gap between largest and smallest parts for sufficiently large $n$. Actually our theorems a
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
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