Let me begin with some preliminary concepts: A positive real-valued function $\varphi: P \rightarrow \Bbb{R}_{>0}$ on a locally finite, ranked poset $(P, \trianglelefteq)$ is *harmonic* if
$\varphi(\emptyset)=1$ and

\begin{equation} \varphi(u)=\sum_{\stackrel{\scriptstyle u \, \triangleleft \, v}{|v| \, = \, |u| + 1}} \varphi(v) \end{equation}

where $\emptyset$ is the unique bottom element of $P$ (which we require to exist) and where $|u|$ denotes the rank of an element $u \in P$. The poset $P$ is 1-*differential* if in addition

$\bullet$ The number of elements covered by both $u$ and $v$ equals the number of elements in $P$ covering both $u$ and $v$ whenever $u \ne v$.

$\bullet$ If $u \in P$ covers exactly $k$ elements then $u$ is covered by exactly $k+1$ elements.

In (https://arxiv.org/abs/math/9712266) Goodman and Kerov introduced a semi-group flow on the space of harmonic functions ${\frak{H}}(P)$ when $P$ is 1-differential. Specifically, for $\tau \in [0,1]$ and $\varphi \in {\frak{H}}(P)$

\begin{equation} C_\tau(\varphi)(v) \, := \, \sum_{k=0}^{|v|} {\tau^k (1-\tau)^{|v|-k} \over {(|v| -k )! }} \sum_{|u| = k} \varphi(u) \dim(u,v) \end{equation}

where $\dim(u,v)$ is the number of saturated chains $p_{|u|} \! \lhd \cdots \lhd p_{|v|}$ in $P$ starting at $p_{|u|} = u$ and ending at $p_{|v|}= v$. A fairly easy calculation reveals that $C_\tau(\varphi)$ is harmonic whenever $\varphi$ is and that $C_\tau (C_\sigma(\varphi)) = C_{\tau \sigma}(\varphi)$. Furthermore we recover the original function $\varphi$ when $\tau = 1$ while we obtain the function

\begin{equation} v \mapsto {1 \over {|v|!}} \dim(\emptyset, v) \end{equation}

when $\tau = 0$, which is known to be harmonic whenever $P$ is 1-differential.

Now consider the Young lattice $(\Bbb{Y}, \subseteq)$ of all integer partitions, ordered by inclusion of their respective Young diagrams. In virtue of the Pieri rule we know that the function

\begin{equation} \varphi(\lambda) \, := \, {s_\lambda({\bf x}) \over {s^n_{\Box}({\bf x})}} \quad \text{where $\lambda \vdash n$} \end{equation}

is a harmonic function on $\Bbb{Y}$ where $s_\lambda({\bf x})$ is the Schur function associated to $\lambda$ and $s_\Box({\bf x}) = x_1 + x_2 + x_3 + \cdots < \infty$ is the Schur function associated to the partition $(1)$. Let's apply the Goodman-Kerov flow to this function: For $\lambda \vdash n$ we get

\begin{equation} C_\tau (\varphi)(\lambda) \, = \, {1 \over {s^n_\Box({\bf x})}} \, \underbrace{\sum_{k=0}^{n} {\tau^k (1-\tau)^{n-k} \over {(n -k )! }} s_\Box^{n-k}({\bf x}) \sum_{|\mu| = k} s_\mu({\bf x}) \dim(\mu,\lambda)}_{\text{call this $s_\lambda({\bf x};\tau)$}} \end{equation}

We may now expand $s_\lambda({\bf x}; \tau)$ as $\sum_{\rho \vdash n} a_{\lambda, \rho}(\tau) s_\rho({\bf x})$.

**Question:** What can be said about the polynomials
$a_{\lambda, \rho}(\tau)$? Have they already been identified/considered in the literature?

**Sub-question:** If the coefficient polynomials $a_{\lambda, \rho}(\tau)$ are (in general) messy, does anything nice happen with $s_\lambda({\bf x}; \tau)$ when we perform either the principal or content specializations, i.e.

\begin{equation} \begin{array}{ll} x_i \mapsto \ \ q^{i-1} \ \ \text{for all $i \geq 1$} & \\ x_i \mapsto \left\{ \begin{array}{ll} 1 &\text{for all $i \leq d$} \\ 0 &\text{for all $i > d$} \end{array} \right. &\text{for some fixed but far out $d \geq 1$} \end{array} \end{equation}

thanks, ines.