Return to Question

thanks, ines.

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 ?

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.

Thanks, ines.

Post-Script: In light of the interpolation question asked in

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?

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

where $\dim(\eta, \lambda)$ is the number of saturated chains in $\Bbb{Y}$ joining $\eta$ and $\lambda$ with $|\eta| \leq |\lambda|$. The existence of this interpolation (which is an special property enjoyed by $1$-differential posets) adds weight to the intuition that there should be some kind of $\tau$-deformation of the Nekrasov-Okounkov formula in the direction of ${\bf x}$.

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 ${\bf x}$.