Combinatorial Interpretations of the Kreweras Triangle in Terms of Subset Tuples

Ange Bigeni

doi:10.37236/7531

Abstract

We show how the combinatorial interpretation of the normalized median Genocchi numbers in terms of multiset tuples, defined by Hetyei in his study of the alternation acyclic tournaments, is bijectively equivalent to previous models like the normalized Dumont permutations or the Dellac configurations, and we extend the interpretation to the Kreweras triangle.

Highlights

For all pair of integers n < m, the set {n, n + 1, . . . , m} is denoted by [n, m], and the set [1, n] by [n]
In his study of the alternation acyclic tournaments [10], Hetyei proved that the median Genocchi number H2n+1 is the number of pairs
One of the results of this paper is to show that the properties of Hetyei’s model extend to the Kreweras triangle, i.e., that #Mn,k = #Mn,k = hn,k for all k ∈ [n]

Summary

Introduction

For all pair of integers n < m, the set {n, n + 1, . . . , m} is denoted by [n, m], and the set [1, n] by [n]. Among the first combinatorial models of the (median) Genocchi numbers [5, 7, 1, 6], there is the set P D2n of the Dumont permutations of the second kind, that is, the permutations σ ∈ S2n+2 such that σ(2i − 1) > 2i − 1 and σ(2i) < 2i for all i ∈ [n + 1], whose cardinality #P D2n equals H2n+1 for all n 0. #P D2Nn,k = #P D2Nn,k = hn,k where the Kreweras triangle (hn,k)n 1,k∈[n] [11] (see Figure 1.1) is defined by h1,1 = 1 and, for all n 2 and k ∈ [3, n], hn,1 = hn−1,1 + hn−1,2 + . The Kreweras triangle appeared recently in the theory of finite type Vassiliev knot invariants [3], more precisely through a polynomial generalization

The Dellac configurations

Hetyei’s model

Feigin’s model

Bijective equivalence with Hetyei’s model