Lah numbers and Lindstr\"om's lemma

TL;DR

This paper offers a combinatorial interpretation of Lah numbers using planar networks and demonstrates, via Lindström's lemma, that the Lah matrix is totally non-negative, revealing new structural properties.

Contribution

It introduces a novel combinatorial interpretation of Lah numbers and establishes the total non-negativity of the Lah matrix through Lindström's lemma.

Findings

Lah numbers can be interpreted combinatorially via planar networks.

The Lah matrix is shown to be totally non-negative.

Lindström's lemma underpins the total non-negativity result.

Abstract

We provide a combinatorial interpretation of Lah numbers by means of planar networks. Henceforth, as a conesquence of Lindstr\"om's lemma, we conclude that the related Lah matrix possesses a remarkable property of total non-negativity.