Generalized Ordered Set Partitions

TL;DR

This paper introduces a generalized form of ordered set partitions with specific constraints, develops its mathematical properties, and explores combinatorial interpretations, extending classical Lah numbers and related structures.

Contribution

It proposes a new generalized sequence of Lah numbers, derives their generating functions, recurrence relations, and identities, and provides combinatorial interpretations for special cases.

Findings

Derived exponential generating function for the generalized Lah numbers

Established recurrence relations and combinatorial identities

Provided combinatorial interpretations for inverse matrices in special cases

Abstract

In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first $r$ elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For this new combinatorial sequence we derive its exponential generating function, some recurrence relations, and combinatorial identities. We prove and present results using combinatorial arguments, generating functions, the symbolic method and Riordan arrays. For some specific cases we provide a combinatorial interpretation for the inverse matrix of the generalized Lah numbers by means of two families of posets.