Barred Preferential Arrangements

A preferential arrangement of a set is a total ordering of the elements of that set with ties allowed. A barred preferential arrangement is one in which the tied blocks of elements are ordered not only amongst themselves but also with respect to one or more bars. We present various combinatorial identities for $r_{m,\ell}$, the number of barred preferential arrangements of $\ell$ elements with $m$ bars, using both algebraic and combinatorial arguments. Our main result is an expression for $r_{m,\ell}$ as a linear combination of the $r_k$ ($= r_{0,k}$, the number of unbarred preferential arrangements of $k$ elements) for $\ell\le k\le\ell+m$. We also enumerate those arrangements in which the sections, into which the blocks are segregated by the bars, must be nonempty. We conclude with an expression of $r_\ell$ as an infinite series that is both convergent and asymptotic.