Abstract
Tiered trees were introduced by Dugan–Glennon–Gunnells–Steingrímsson as a generalization of intransitive trees that were introduced by Postnikov, the latter of which have exactly two tiers. Tiered trees arise naturally in counting the absolutely indecomposable representations of certain quivers, and enumerating torus orbits on certain homogeneous varieties over finite fields. By employing generating function arguments and geometric results, Dugan et al. derived an elegant formula concerning the enumeration of tiered trees, which is a generalization of Postnikov’s formula for intransitive trees. In this paper, we provide a bijective proof of this formula by establishing a bijection between tiered trees and certain rooted labeled trees. As an application, our bijection also enables us to derive a refinement of the enumeration of tiered trees with respect to level of the node 1.
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