Cross Recurrence Relations for $r$-Lah Numbers

Abstract

We give a new combinatorial interpretation of Lah and $r$-Lah numbers.
We establish two cross recurrence relations: the first one, which uses
an algebraic approach, is a recurrence relation of order two with
rational coefficients; the second one uses a combinatorial proof and
is a recurrence relation with integer coefficients. We also express
$r$-Lah numbers in terms of Lah numbers. Finally, we give identities
related to rising and falling factorial powers.