  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.
