Matrices Related to the Idempotent Numbers and the Numbers of Planted Forests

Abstract

In this paper, we study the matrices related to the idempotent number and the number of planted forests with \(k\) components on the vertex set \([n]\). As a result, the factorizations of these two matrices are obtained. Furthermore, the discussion goes to the generalized case. Some identities and recurrences involving these two special sequences are also derived from the corresponding matrix representations.