Boolean complexes for Ferrers graphs
Abstract
In this paper we provide an explicit formula for calculating the boolean number of a Ferrers graph. By previous work of the last two authors, this determines the homotopy type of the boolean complex of the graph. Specializing to staircase shapes, we show that the boolean numbers of the associated Ferrers graphs are the Genocchi numbers of the second kind, and obtain a relation between the LegendreStirling numbers and the Genocchi numbers of the second kind. In another application, we compute the boolean number of a complete bipartite graph, corresponding to a rectangular Ferrers shape, which is expressed in terms of the Stirling numbers of the second kind. Finally, we analyze the complexity of calculating the boolean number of a Ferrers graph using these results and show that it is a significant improvement over calculating by edge recursion.
 Publication:

arXiv eprints
 Pub Date:
 August 2008
 arXiv:
 arXiv:0808.2307
 Bibcode:
 2008arXiv0808.2307C
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Topology;
 05A15;
 55P15;
 05C99;
 05A19
 EPrint:
 final version, to appear in the The Australasian Journal of Combinatorics