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

TL;DR

This paper explores a family of generating functions related to preferential arrangements, introducing restricted barred arrangements and deriving new identities with combinatorial proofs for all positive integer parameters.

Contribution

It generalizes existing generating functions for preferential arrangements by introducing a broader family with combinatorial structures and identities, extending prior work by Nelsen and Schmidt.

Findings

Derived new identities for restricted barred preferential arrangements.

Proposed a generalized family of generating functions $P^{r}_{j}(m)$.

Provided combinatorial proofs and conjectures on arrangement counts.

Abstract

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 this study we propose combinatorial structures whose integer sequences are generated by members of the family for all values of $k$ in $\mathbb{Z}^+$. To do this we use a notion of restricted barred preferential arrangements. We then propose a more general family of generating functions $P^{r}_{j}(m)=\frac{e^{rm}}{(2-e^m)^j}$ for $r,j\in\mathbb{Z^+}$. We derive some new identities on restricted barred preferential arrangements and give their combinatorial proofs. We also propose conjectures on number of restricted barred preferential arrangements.