Inequalities among two rowed immanants of the q-Laplacian of trees and odd height peaks in generalized Dyck paths

Abstract

Let T be a tree on n vertices and let ${\mathcal{L}}_q^T$ be the q-analogue of its Laplacian. For a partition $\lambda ⊢n$, let the normalized immanant of ${\mathcal{L}}_q^T$ indexed by λ be denoted as ${\overline{\mathrm{I}\mathrm{m}\mathrm{m}}}_{\lambda }\left({\mathcal{L}}_q^T\right)$. A string of inequalities among ${\overline{\mathrm{I}\mathrm{m}\mathrm{m}}}_{\lambda }\left({\mathcal{L}}_q^T\right)$ 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 ${\mathfrak{S}}_n$ 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 ${\overline{\mathrm{I}\mathrm{m}\mathrm{m}}}_{{\lambda }_1}\left({\mathcal{L}}_q^{T_1}\right)$ and ${\overline{\mathrm{I}\mathrm{m}\mathrm{m}}}_{{\lambda }_2}\left({\mathcal{L}}_q^{T_2}\right)$ when $T_1$ and $T_2$ are comparable trees in the ${\mathrm{G}\mathrm{T}\mathrm{S}}_n$ poset and when ${\lambda }_1$ and ${\lambda }_2$ are both two rowed partitions of n, with ${\lambda }_1$ having a larger first part than ${\lambda }_2$.

Keywords:

• Normalized immanant
• q-Laplacian matrix
• tree
• Dyck paths
• Riordan paths

2020 Mathematics Subject Classifications:

• 05C05
• 05A19
• 15A15

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.

Additional information

Funding

The first author would like to acknowledge SERB, Government of India for providing a National Postdoctoral fellowship with file number PDF/2018/000828. The last author acknowledges support from project SERB/F/252/2019-2020 given by the Science and Engineering Research Board (SERB), India.