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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 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.