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The B n (k) poly-Bernoulli numbers — a natural generalization of classical Bernoulli numbers (B n = Bn (1)) — were introduced by Kaneko in 1997. When the parameter k is negative then B n (k) is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that B n (−k) counts the so called lonesum 0–1 matrices of size n × k. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combinatorial explanation of Kaneko’s recursive formula for poly-Bernoulli numbers.
We study a combinatorial notion where given a set S of lattice points one takes the set of all sums of p distinct points in S, and we ask the question: ‘if S is the set of lattice points of a convex lattice polytope, is the resulting set also the set of lattice points of a convex lattice polytope?’ We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.
Abstract
The inclusion-exclusion principle is one of the basic theorems in combinatorics. In this paper the inclusion-exclusion principle for IF-sets on generalized probability measures is studied. The basic theorems are proved.
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A special case of a conjecture of Ryser states that if a 3-partite 3-uniform hypergraph has at mostv pairwise disjoint edges then there is a set of vertices of cardinality at most 2v meeting all edges of the hypergraph. The best known upper bound for the size of such a set is (8/3)v, given by Tuza [7]. In this note we improve this to (5/2)v.
