Acyclic orientations and poly-Bernoulli numbers

Abstract

In 1997, Masanobu Kaneko defined poly-Bernoulli numbers, which bear much the same relation to polylogarithms as Berunoulli numbers do to logarithms. In 2008, Chet Brewbaker described a counting problem whose solution can be identified with the poly-Bernoulli numbers with negative index, the lonesum matrices.The main aim of this paper is to give formulae for the number of acyclic orientations of a complete bipartite graph, or of a complete bipartite graph with one edge added or removed.Our formula shows that the number of acyclic orientations of K n 1 ,n 2 is equal to the poly-Bernoulli number B (−n 2 ) n 1. We also give a simple bijective identification of acyclic orientations and lonesum matrices.We make some remarks on the context of our result, which are expanded in another paper.

On the Number of Acyclic Orientations of Complete $k$-Partite Graphs

“…The algorithm we have presented is based on a recurrence derived from reduction to counting the number of Hamiltonian paths in complete kpartite graphs. Our result extends previous work by Cameron et al [3], although we are unable to give a similarly neat formula when k > 2. A natural future extension is to consider what happens if we remove an edge from a complete k-partite graph, and how to adapt our algorithm to that resulting graph.…”

“…We let Φ = F ∈F S F . We prove the following observation, which is a generalisation of Cameron et al's argument for complete bipartite graphs in [3], and is implicit in Reidy's representation of acyclic orientations in [7].…”

“…For the case where k = 2, we get a complete bipartite graph, and Cameron et al's result in [3] applies. Let n denote the total number of vertices in G n : n = k i=1 n i .…”

“…Observe that π can be viewed as a sequence of maximal blocks such that each block consists only of vertices that all come from the same vertex part V i , and the blocks are maximal in the sense that we do not have two blocks with vertices from the same V i sitting next to each other in π: we would consider that as one block. As noted in [7] and [3], if we are to change the order of the vertices within a single such block of π, we would still get the same σ ∈ AO(G n ), because there are no edges between any two vertices lying in the same block, hence, no edge would be changing its orientation. However, if we change the order of the blocks themselves, and/or we change the constitution of the blocks (i.e., which vertices go in which blocks), we obtain a different acyclic orientation.…”

“…Our work follows from Cameron et al's [3] on complete bipartite graphs K n 1 ,n 2 , where n 1 , n 2 are the sizes of the two vertex parts. In [3] Cameron et al give an expression for the number of acyclic orientations of K n 1 ,n 2 : namely, that…”