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Inequalities among two rowed immanants of the q-Laplacian of trees and odd height peaks in generalized Dyck paths
Journal
Journal of Difference Equations and Applications
ISSN
10236198
Date Issued
2022-01-01
Author(s)
Nagar, Mukesh Kumar
Lal, Arbind Kumar
Sivasubramanian, Sivaramakrishnan
Abstract
Let T be a tree on n vertices and let (Formula presented.) be the q-analogue of its Laplacian. For a partition (Formula presented.), let the normalized immanant of (Formula presented.) indexed by λ be denoted as (Formula presented.). A string of inequalities among (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) and (Formula presented.) when (Formula presented.) and (Formula presented.) are comparable trees in the (Formula presented.) poset and when (Formula presented.) and (Formula presented.) are both two rowed partitions of n, with (Formula presented.) having a larger first part than (Formula presented.).
Volume
28
Publication link
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