Unfair distributions counted by the generalized Stirling numbers
MTMT : 33067781
Megjelenés dátuma : 2022
Folyóirat címe : Integers: Electronic Journal of Combinatorial Number Theory
Évfolyam : 22
Oldalszám : 1-28
Dokumentum típusa : folyóiratcikk
Kulcsszó : Stirling numbers, Bell numbers, combinatory, sign-changing involution, direct bijection, Természettudományok, Természettudományok/Matematika- és számítástudományok
Absztrakt :
We investigate the generalized Stirling numbers S(n, k; α, β, γ) introduced by Hsu
and Shiue from a combinatorial point of view. We present a combinatorial interpretation in terms of certain restricted distributions of labeled balls into unlabeled
cells and a special cell where all cells are divided into distinct compartments. Using our interpretation, we find combinatorial proofs of several identities involving
S(n, k; α, β, γ) and the associated generalized Bell numbers. Connections are made
with some prior combinatorial models for the r-Lah numbers and other arrays, one
via a sign-changing involution and another through a direct bijection. Finally, an
additional parameter is introduced into our model which allows for further generalization.