Combinatorial interpretations of the Jacobi-Stirling numbers

Yoann Gelineau (1) , Jiang Zeng (1)

1 ICJ - Institut Camille Jordan

The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds.

Mots clés: Jacobi-Stirling numbers; Legendre-Stirling numbers; Stirling numbers; central factorial numbers; signed partitions; quasi-permutations; simply hooked quasi-permutations; Riordan complexes

Domaines: Combinatoire [math.CO]

Yoann Gelineau, Jiang Zeng. Combinatorial interpretations of the Jacobi-Stirling numbers. 2009. ⟨hal-00385228⟩