  Let $T$ be a tree on $n$ vertices and let $L_q^T$ be the $q$-analogue of its
Laplacian. For a partition $\lambda \vdash n$, 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 $GTS_n$ poset and when $\lambda_1$ and $\lambda_2$
are both two rowed partitions of $n$, with $\lambda_1$ having a larger first
part than $\lambda_2$.
