10 research outputs found
The largest singletons of set partitions
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 denote the number of partitions of
with the largest singleton for .
In this paper, several explicit formulas for , involving a
Dobinski-type analog, are obtained by algebraic and combinatorial methods, many
combinatorial identities involving and Bell numbers are presented by
operator methods, and congruence properties of are also investigated.
It will been showed that the sequences and
(mod ) are periodic for any prime , and contain a
string of consecutive zeroes. Moreover their minimum periods are
conjectured to be for any prime .Comment: 14page
The largest singletons in weighted set partitions and its applications
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 denote the total
weight of partitions on with the largest singleton . In this
paper, explicit formulas for and many combinatorial
identities involving 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
is the -th derangement number or the number of permutations of
with no fixed points.Comment: 15page
Topics in Graph Compositions
For any discrete undirected graph G with vertex set V(G) and edge set E(G) (respectively), a graph composition of G is defined to be a partition of V(G) where every element of the partition yields a connected, induced subgraph of G. This dissertation is comprised of 5 chapters. The first is a general introduction to the concept of graph compositions and a survey of previously researched work; the second focuses on the composition number of deletions of specific graphs from complete graphs; the third focuses on establishing bounds for the composition number of general graphs and the Bell number coefficients of general graphs; the fourth focuses on the connection between graph compositions and Aitken\u27s array, a well researched array; finally, the fifth focuses on the number of compositions of graphs where the number of components is restricted
Staircase Packings of Integer Partitions
An integer partition is a weakly decreasing sequence of positive integers. We study the family of packings of integer partitions in the triangular array of size n, where successive partitions in the packings are separated by at least one zero. We prove that these are enumerated by the Bell-Like number sequence (OEIS A091768), and investigate its many recursive properties. We also explore their poset (partially ordered set) structure. Finally, we characterize various subfamilies of these staircase packings, including one restriction that connects back to the original patterns of the whole family