Genocchi Numbers and f-Vectors of Simplicial Balls

The Genocchi numbers are the coefficients of the generating function 2t/(e^t+1). In this note we will give an equation for simplicial balls which involves this numbers. It relates the number of faces in the interior of the ball with the number of faces in the boundary of the ball. This is a variation of similar equations given in [McMullen 2004] and [Herrmann and Joswig 2006].Comment: revised version, to appear in European Journal of Combinatorics; minor change

Combinatorial proofs of some properties of tangent and Genocchi numbers

The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger divisibility property is known in the studies of Bernoulli and Genocchi numbers, namely, the divisibility of $(n+1)T_{2n+1}$ by $2^{2n}$. The traditional proofs of this fact need significant calculations. In the present paper, we provide a combinatorial proof of the latter divisibility by using the hook length formula for trees. Furthermore, our method is extended to $k$-ary trees, leading to a new generalization of the Genocchi numbers

A bijection between the irreducible k-shapes and the surjective pistols of height k-1

This paper constructs a bijection between irreducible $k$-shapes and surjective pistols of height $k-1$, which carries the "free $k$-sites" to the fixed points of surjective pistols. The bijection confirms a conjecture of Hivert and Mallet (FPSAC 2011) that the number of irreducible $k$-shape is counted by the Genocchi number $G_{2k}$