Geodesic
Some remarks on the joint distribution of descents and inverse descents
Mirkó Visontai1
1Royal Institute of Technology
The electronic journal of combinatorics, Tome 20 (2013) no. 1

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Zbl arXiv
We study the joint distribution of descents and inverse descents over the set of permutations of $n$ letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that give the recurrence for these generating functions. As a result we devise a recurrence for the coefficients in question but are unable to settle the conjecture. We examine generalizations of the conjecture and obtain a type $B$ analog of the recurrence satisfied by the two-variable generating function. We also exhibit some connections to cyclic descents and cyclic inverse descents. Finally, we propose a combinatorial model for the joint distribution of descents and inverse descents in terms of statistics on inversion sequences.
DOI : 10.37236/2135
Classification : 05A05, 05A15
Mots-clés : permutations, descents, inverse descents, Eulerian polynomials

Mirkó Visontai  1

1 Royal Institute of Technology 
Mirkó Visontai. Some remarks on the joint distribution of descents and inverse descents. The electronic journal of combinatorics, Tome 20 (2013) no. 1. doi: 10.37236/2135
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