A Combinatorial Analysis Of Higher Order Generalised Geometric Polynomials: A Generalisation Of Barred Preferential Arrangements

Generalised Barred Preferential Arrangements

A barred preferential arrangement is a preferential arrangement onto which a number of identical bars are inserted in between the blocks of the preferential arrangement. In this study we examine combinatorial properties of barred preferential arrangements whose elements are colored with a number of available colors.

On some properties and relations between restricted barred preferential arrangements, multi-poly-Bernoulli numbers and related numbers

The introduction of bars in-between blocks of an ordered set partition(preferential arrangement) results in a barred ordered set partition(barred preferential arrange- ment). Having the restriction that some blocks of barred preferential arrangements to have a maximum of one block results in restricted barred preferential arrange- ments. In this study we establish relations between number of restricted barred preferential arrangements, multi-poly-Bernoulli numbers and numbers related to multi-poly-Bernoulli numbers. We prove a periodicity property satisfied by multi- poly-Bernoulli numbers hav…

On Some Identities of Barred Preferential Arrangements

A preferential arrangement of a finite set is an ordered partition. Associated with each such ordered partition is a chain of subsets or blocks endowed with a linear order. The chain may be split into sections by the introduction of a vertical bar, leading to the notion of a barred preferential arrangements. In this paper we derive some combinatorial identities satisfied by the number of possible barred preferential arrangements of an $n$-element set. We illustrate with some suitable examples highlighting some important consequences of the identities.

A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements

A preferential arrangement of a set $X_n=\{1,2,...,n\}$ is an ordered partition of the set $X_n$ induced with a linear order. Separation of blocks of a preferential arrangement with bars result in the notation of barred preferential arrangements. Roger Nelsen and Harvey Schmidt have proposed the family of generating functions $P^k(m)=\frac{e^{km}}{2-e^m}$; which for $k=0$ and for $k=2$ they have shown that the generating functions are exponential generating functions for the number of preferential arrangements of a set $X_n$ and the number of chains in the power set of $X_n$ respectively. In t…

Barred Preferential Arrangements

A preferential arrangement of a set is a total ordering of the elements of that set with ties allowed. A barred preferential arrangement is one in which the tied blocks of elements are ordered not only amongst themselves but also with respect to one or more bars. We present various combinatorial identities for r_{m,l}, the number of barred preferential arrangements of l elements with m bars, using both algebraic and combinatorial arguments. Our main result is an expression for r_{m,l} as a linear combination of the r_k (= r_{0,k}, the number of unbarred preferential arrangements of k elements)…

高阶 r-Dowling 多项式

Higher Order r-Dowling polynomials

本研究利用带竖线的偏好排列，为高阶 r-Dowling 多项式提供组合解释，并证明组合恒等式、积分表示及渐近结果，展示其与 Bell 多项式的联系。