Buried in the physics paper by Nekrasov and Okounkov, a strange identity is proven: $$ \prod_{n > 0} (1 - q^n)^{\mu^2-1} = \sum_{\mathbf{k}} q^{|\mathbf{k}|} \prod_{\square \in k} \left( 1 - \frac{\mu^2}{h(\square)^2}\right) $$ where the left side is a q-series and the right side is the sum over all partitions. Ihis was proven by physical considerations, evaluating the Yang-Mills partition function in 2 different ways.

The partitions could index representations of the permutation group $S_n$. We can define measure on partitions, $\mathrm{Irr}(S_n)$ by

$$ \mathbb{P}\_{\mu, t} (\mathbf k) = \prod_{n \geq 1} (1-t^n)^{1-\mu^2} q^{|\mathbf{k}|} \prod_{\square \in k} \left( 1 - \frac{\mu^2}{h(\square)^2}\right) $$

In fact, 3 years later Alexei Borodin explains this formula interpolates between uniform and Plancherel measures on partitions.

Can this be extended to a q,t-deformation of uniform measure on the permutation group? Maybe through something similar to Robinson-Schensted correspondence.