This post is related to the issues addressed in
A q,t-extension of Plancherel Measure thru Yang-Mills Theory ?
however the generalization/interpolation which John Mangual asks for looks different from the one proposed here.
In what follows $\lambda$ will be an integer partition (of size $|\lambda| = n$), the notation $b \in \lambda$ will indicate that $b$ is a cell or box in the Young diagram of $\lambda$, and $\mathrm{h}(b)$ will denote the hook length of the hook determined by $b$ in $\lambda$'s Young diagram. Furthermore $\dim(\lambda)$ will denote the number of saturated chains $\lambda^{(0)} \subset \cdots \subset \lambda^{(n)}$ in the Young lattice $\Bbb{Y}$ beginning at the empty partition $\lambda^{(0)} = \emptyset$ and ending at $\lambda^{(n)} = \lambda$. Equivalently $\dim(\lambda)$ is the dimension of the corresponding irreducible representation $V_\lambda$ of the symmetric group $S_n$.
The Nekrasov-Okounkov formula is the following $t$-deformation of the classical identity for the partition function, namely:
\begin{equation} \begin{array}{ll} \displaystyle \sum_{\lambda \in \Bbb{Y}} z^{|\lambda|} \,\prod_{b \in \lambda} \, \Big(1 - {t \over {\mathrm{h}^2(b})} \Big) &\displaystyle = \ \sum_{n \geq 0} \, {z^n \over {n !}} \, \sum_{|\lambda| = n} \, {\dim^2(\lambda) \over {n!}} \, \prod_{b \in \lambda} \, \big( \mathrm{h}^2(b) - t \big) \\ \\ &\displaystyle \stackrel{!}{=} \ \prod_{k \geq 1} \, \big(1 - z^k \big)^{t-1} \end{array} \end{equation}
with the specialization $t=0$ corresponding to the well known identity for the generating function of $p(n)$, the number of integer partitions of $n \geq 0$:
\begin{equation} \sum_{n \geq 0} \, p(n) \, z^n = \prod_{k \geq 1} \, \big( 1 - z^k \big)^{-1} \end{equation}
My first question concerns the internal sum
\begin{equation} \sum_{|\lambda| = n} \, {\dim^2(\lambda) \over {n!}} \, \prod_{b \in \lambda} \, \big( \mathrm{h}^2(b) - t \big) \end{equation}
which we can view as an expectation value for the $t$-statistic $H_t(\lambda) = \prod_{b \in \lambda} \, \big(\mathrm{h}^2(b) - t \big) $ with respect to the Plancherel measure $\mu^{(n)}_\mathrm{P}(\lambda) = {1 \over {n!}} \dim^2(\lambda)$ on the set of partitions of size $|\lambda| =n$. Let's denote this expectation value as $\langle H_t \rangle_{\mathrm{P},n}$. Clearly it is a polynomial in $t$.
Question 1. Is there a nice (e.g. combinatorial) formula describing how $\langle H_t \rangle_{\mathrm{P},n}$ factorizes as a polynomial in $t$? Indeed, should it factorize nicely?
Instead of using the (family of) Plancherel measure(s) $\mu^{(n)}_\mathrm{P}$ we may use a coherent, ergodic family of measures $\mu^{(n)}_\varphi$ associated to another choice of normalised, minimal, harmonic function $\varphi: \Bbb{Y} \longrightarrow \Bbb{R}_{>0}$. Recall that the measure $\mu_\varphi^{(n)}$ on the set of partitions $\lambda$ of size $|\lambda| = n$ is defined by
\begin{equation} \mu_\varphi^{(n)} (\lambda) := \ \dim(\lambda) \, \varphi(\lambda) \end{equation}
Among the supply of minimal, normalised, harmonic functions are the Schur-states $\varphi_{\bf x}$ defined by $\varphi_{\bf x}(\lambda) = {\scriptstyle \frak{S}}_\lambda(x_1, x_2, x_3, \dots)$ where ${\scriptstyle \frak{S}}_\lambda$ is the Schur function associated to $\lambda \in \Bbb{Y}$ and where ${\bf x} = (x_1, x_2, x_3, \dots)$ is any sequence of positive real numbers with $0 \leq x_k \leq 1$ and $x_{k+1} \leq x_k$ for all $k \geq 1$ and with convergent sum $\sum_{k \geq 1} x_k = 1$. Accordingly define the corresponding Schur measure(s) by $\mu^{(n)}_{\bf x}(\lambda) = \dim(\lambda) \, {\scriptstyle \frak{S}}_\lambda(x_1, x_2, x_3, \dots)$.
Question 2. Is a version of the Nekrasov-Okounkov formula known where the role of the Plancherel measure $\mu^{(n)}_\mathrm{P}$ is replaced by a Schur measure $\mu^{(n)}_{\bf x}$? Specifically an identity for \begin{equation} \sum_{n \geq 0} \, {z^n \over {n !}} \, \sum_{|\lambda| = n} \, \mu^{(n)}_\mathrm{x} \, \prod_{b \in \lambda} \, \big( \mathrm{h}^2(b) - t \big) \ = \ ? \end{equation}
It's not clear to me what the $t=0$ version would be; related perhaps to some kind of content identity.
Post-Script. In light of the interpolation question asked in
A q,t-extension of Plancherel Measure thru Yang-Mills Theory ?
I would add that there is a (family of) coherent, ergodic measures $\mu_{{\bf x}, \tau}^{(n)}$ interpolating between the Plancherel measure(s) $\mu^{(n)}_\mathrm{P}$ and the Schur measures $\mu^{(n)}_{\bf x}$ for any choice of (admissible) real sequence ${\bf x}=(x_1, x_2, x_3, \dots)$. The normalised, non-negative, minimal, harmonic function $\varphi_{{\bf x},\tau}$ associated to the family $\mu_{{\bf x}, \tau}^{(n)}$ is given by
\begin{equation} \varphi_{{\bf x},\tau}(\lambda) \ = \ \sum_{k = 0}^n \, {\tau^k (1-\tau)^{n-k} \over {(n-k)!}} \, \sum_{|\eta| = k} \, \varphi_{\bf x}(\eta) \dim(\eta, \lambda) \end{equation}
where $\dim(\eta, \lambda)$ is the number of saturated chains in $\Bbb{Y}$ joining $\eta$ and $\lambda$ with $|\eta| \leq |\lambda|$. Clearly $\varphi_{{\bf x},1} = \varphi_{\bf x}$ and $\varphi_{{\bf x},0} = \varphi_\mathrm{P}$ where $\varphi_\mathrm{P}(\lambda) = {1 \over {n!}} \dim(\lambda)$ is the Plancherel state. The existence of this interpolation (which is an special property enjoyed by $1$-differential posets and discovered by Goodman and Kerov) adds weight to the intuition that there should be some kind of $\tau$-deformation of the Nekrasov-Okounkov formula in the direction of ${\bf x}$.
thanks, ines.
