Inequalities among two rowed immanants of the $q$-Laplacian of Trees and Odd height peaks in generalized Dyck paths

Let $T$ be a tree on $n$ vertices and let $L_q^T$ be the $q$-analogue of its Laplacian. For a partition $\lambda ⊢n$, let the normalized immanant of $L_q^T$ indexed by $\lambda$ be denoted as $d_{\lambda }(L_q^T)$. A string of inequalities among $d_{\lambda }(L_q^T)$ is known when $\lambda$ varies over hook partitions of $n$ as the size of the first part of $\lambda$ decreases. In this work, we show a similar sequence of inequalities when $\lambda$ varies over two row partitions of $n$ as the size of the first part of $\lambda$ decreases. Our main lemma is an identity involving binomial coefficients and irreducible character values of $S_n$ 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 $d_{{\lambda }_1}(L_q^{T_1})$ and $d_{{\lambda }_2}(L_q^{T_2})$ when $T_1$ and $T_2$ are comparable trees in the $GT{S}_n$ poset and when ${\lambda }_1$ and ${\lambda }_2$ are both two rowed partitions of $n$, with ${\lambda }_1$ having a larger first part than ${\lambda }_2$.