 Open Access Publications from the University of California

## Dowling Set Partitions, and Positional Marked Patterns

• Author(s): Thamrongpairoj, Sittipong
• et al.
Abstract

This dissertation consists of two parts. First, we define and study positional marked patterns. A positional marked pattern tau is a permutation where one of elements in tau is underlined. Given a permutation sigma, we say that sigma_i is tau-match at position i if tau occurs in sigma in such a way that sigma_i plays the role of the underlined element in the occurrence. We let pmp_tau(sigma) denote the number of position i which sigma is tau-match. This defines a new class of statistics on permutations, where we study such statistics and prove a number of results. In particular, we prove that two positional marked patterns (1)23 and (1)32 give rise to two statistics that have the same distribution. The equidistibution phenomenon also occurs in other several pairs of patterns like 1(2)3 and 1(3)2, which we prove in this dissertation. The second part of the dissertation focuses on the Whitney numbers of Dowling lattices. In the papers [2, 3], Benoumhani defined two polynomials Fm,n,1(x)

and Fm,n,2(x). Then, he defined Am(n,k) and Bm(n,k) to be the polynomials satisfying

Fm,n,1(x) = Sum_k=0^n Am(n,k)x^(n-k)(x+1)^k

and

Fm,n,2(x) = Sum_k=0^n Bm(n,k)x^(n-k)(x+1)^k.

In this dissertation, we give a combinatorial interpretation of coefficients of Am+1(n,k) and prove a symmetry of the coefficients, namely Am+1(n,k)|m^s = Am+1(n,n-k)|m^(n-s). We also give a combinatorial interpretation of Bm+1(n,k), prove that Bm+1(n,n-1) is a polynomial in m with non-negative integer coefficients, and prove that Bm+1(n,n-2) is a polynomial in m with non-negative integer coefficients except for the coefficient of m^(n-1) which is -(n-1) for n >=6.