Abstract
Let T be a tree on n vertices and let be the q-analogue of its Laplacian. For a partition , let the normalized immanant of indexed by λ be denoted as . A string of inequalities among is known when λ varies over hook partitions of n as the size of the first part of λ decreases. In this work, we show a similar sequence of inequalities when λ varies over two row partitions of n as the size of the first part of λ decreases. Our main lemma is an identity involving binomial coefficients and irreducible character values of indexed by two row partitions. Our proof can be interpreted using the combinatorics of Riordan paths and our main lemma admits a nice probabilisitic interpretation involving peaks at odd heights in generalized Dyck paths or equivalently involving special descents in Standard Young Tableaux with two rows. As a corollary, we also get inequalities between and when and are comparable trees in the poset and when and are both two rowed partitions of n, with having a larger first part than .
Acknowledgments
The authors thank the anonymous referees for their careful reading of our manuscript and their many insightful comments and suggestions that improved the presentation of this paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Correction Statement
This article has been corrected with minor changes. These changes do not impact the academic content of the article.