The Largest Singletons In Weighted Set Partitions

In2010, Sun and Wu studied the number An,k of partitions of [n+1] with the largest singleton k+1, obtained several explicit formulas for Ank, involving a Dobinski-type analog, present many combinatorial identities involving Ank and Bell numbers and investigated congruence properties of An,k. In this thesis, we further to extend the partial work of Sun and Wu, namely, to concentrate on the problem of the largest singletons in weighted set partition. We mainly focus on the relationship between An,k(t) with different weights and some classical combinatorial sequences by using algebraic method, combinatorial interpretations and umbral operators. The principal work includes the following three aspects:1) In the case when t=(0!,1!,2!,…), each block of partitions is endowed a cycle structure, such partitions are equivalent to permutations. As a simple application, we not only re-derive the relation between the derangement numbers Dn and the sum of the values of the largest fixed points of all non-derangements which is originally obtained by Deutsch and Elizalde, but also prove a classical relation between DH and enumeration of rooted labeled trees by using Riordan identity.2) In the case when t1=t2=1and t1=0for (j>3), we study partition of[n+1] with no blocks of sizes greater than2, such partitions are equivalent to involutions. We deduce some identities related to the number Qn,k of involutions of [n+1] with the largest fixed point k+1.3) In the case when tj=jj-1(j>1), each block of partitions of [n+1] is assigned a rooted labeled tree structure. Such partitions are equivalent to rooted labeled forest with the largest singleton tree labeled by k+1. We also provide some identities related to the number Ln,k of rooted labeled forest on [n+1] with the largest singleton tree labeled by k+1.