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I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls the Nekrasov-Okounkov hook length formula. It says

$$\prod_{n=1}^{\infty}(1-x^n)^{\beta-1}=\sum_n\sum_{\lambda \vdash n}\sum_{v\in H(\lambda)}\big(1-\frac{\beta}{h_v^2}\big)x^{|\lambda|},$$

where $h_v$ is the hook length of the box $v$ in the Young tableau of $\lambda$.

This formula is remarkable since taking special values of $\beta$ yields many classical results like

  • the Euler Pentagonal number theorem

$$\prod_{n=1}^{\infty}(1-q^n)=\sum_{n=-\infty}^{\infty}(-1)^nq^{\frac{3n^2-n}{2}},$$

  • the Jacobi triple product formula

$$\prod_{n=1}^{\infty}(1-q^n)^3=\sum_{n\geq 1}(-1)^n(2n+1)q^{n(n+1)/2},$$

  • the Ramanujan $\tau$ function

$$\tau(n)=\sum_{\lambda\vdash n-1}\sum_{v\in H(\lambda)}\big(1-\frac{25}{h_v^2}\big).$$

My questions are:

1) Can we prove the huge list of congruence relations of the $\tau$ function in terms of the above formula?

2) By multiplying with certain powers of $q$, one obtains powers of the Dedekind eta function. Can we have new insights of the modularity in terms of properties of the symmetric group?

Sorry that my second question is vague.

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