  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.
