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A preferential arrangement of a set $X_n=\{1,2,...,n\}$ is an orderedpartition of the set $X_n$ induced with a linear order. Separation of blocks ofa preferential arrangement with bars result in the notation of barredpreferential arrangements. Roger Nelsen and Harvey Schmidt have proposed thefamily 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 exponentialgenerating 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 studywe propose combinatorial structures whose integer sequences are generated bymembers of the family for all values of $k$ in $\mathbb{Z}^+$. To do this weuse a notion of restricted barred preferential arrangements. We then propose amore general family of generating functions  $P^{r}_{j}(m)=\frac{e^{rm}}{(2-e^m)^j}$ for $r,j\in\mathbb{Z^+}$. We derivesome new identities on restricted barred preferential arrangements and givetheir combinatorial proofs. We also propose conjectures on number of restrictedbarred preferential arrangements.

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