Acyclic orientations and poly-Bernoulli numbers

Abstract

The main contribution of this paper is a formula for the number of acyclic orientations of a complete bipartite, $K_{n_1,n_2},$ revealing that it is equal to the poly-Bernoulli number $B_{n_1}^{(-n_2)}$ introduced in 1997 by Kaneko. We also give a simple bijective identification of acyclic orientations and lonesum matrices, confirming a 2008 result of Brewbaker, and show that the poly-Bernoulli numbers behave concavely, as well being symmetric. A second goal is to explore the behaviour of more general complete $r$-partite graphs in the space of the number of acyclic orientations. We prove that the number of acyclic orientations of complete bipartite graphs on $n$ vertices is a unimodular concave function maximised by the Tur\'an graph $T(2,n).$ For tripartite graphs, we derive an explicit formula for its number of its acyclic orientations, for the specific case with a single vertex partition. For more complex complete $r$-partite graphs an algorithmic approach is suggested, taking advantage of the relationship with graph colouring through the result of Stanley, namely the value of the graph colouring polynomial at the value -1. An underlying theme is the exploration of the space of the number of acyclic orientations of graphs. We present various theoretical and computational results, with additional, conjectures for both complete $r$-partite and general graphs to guide future research.