Barred Preferential Arrangements

TL;DR

This paper explores the combinatorial properties of barred preferential arrangements, providing identities and formulas for counting arrangements with various constraints, including nonempty sections and asymptotic series.

Contribution

It introduces new identities and an explicit formula for counting barred preferential arrangements, extending understanding of arrangements with multiple bars and nonempty sections.

Findings

Derived combinatorial identities for r_{m,l}

Expressed r_{m,l} as a linear combination of unbarred arrangements

Presented an infinite series for r_l with convergence and asymptotic properties

Abstract

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,l}, the number of barred preferential arrangements of l elements with m bars, using both algebraic and combinatorial arguments. Our main result is an expression for r_{m,l} as a linear combination of the r_k (= r_{0,k}, the number of unbarred preferential arrangements of k elements) for l <= k<=l+m. We also study 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_l as an infinite series that is both convergent and asymptotic.