Powers of Euler Products and Han’s Marked Hook Formula

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I heard from Dan Romik and Richard Stanley about a very exciting development in enumerative combinatorics. It is quite amazing how new uncharted sections of the gold mine of tableaux, hooks, partitions, and permutations are repeatedly being discovered. Guo-Niu HAN proved the following:

\sum f^2_{\lambda} \sum h^2= n! n(3n-1)/2.

Here, the first sum runs over all partitions \lambda of n, f_{\lambda} is the number of standard young tableaux corresponding to \lambda, and the second sum runs over all hook lengths of the Ferrers diagram corresponding to \lambda. Of course, this formula reminds us of the classical result \sum f^2_{\lambda} = n!, and of the hook formula for the value of f_{\lambda} itself. 

Han’s result is one of many exciting other applications of a recent remarkable formula for Euler’s product (1-x) (1-x^2) \cdots (1-x^k) \cdots raised to an arbitrary complex power z-1, via an expansion based on tableaux. This later formula was first discovered by Nekrasov and Okounkov in the context of Seiberg-Witten theory, and was later rediscovered by Han himself who provided another proof, and presented many remarkable applications. It also gives new proofs and new perspectives to several classical formulas regarding powers of Euler products going back to Euler himself, Jacobi, Ramanujan, and more recently, Macdonald, Kostant and many others.

Added July 11, long overdue: Dan showed me the actual general Nekrasov-Okounkov’s formula, hints regarding Han’s proof, and how the marked hook formula is derived. The general formula for powers of Euler products goes like this: \prod_{n=1}^{\infty}(1-x^n)^{\beta -1} = \sum _{n=0}^{\infty} (\sum _{\lambda \vdash n} \prod _{c \in \lambda} (1-\beta/h_c^2(\lambda ))x^n. The proof relies on a formula of Macdonald for the case that \beta=t^2 and t is an odd integer. Macdonald found a beautiful expansion in this case of the form  \sum _{(n_1,n_2,\dots,n_t)\in V_t}x^{(n_1^2+n_2^2+ \dots +n_t^2)/2- 1/24}, where  V_t referred to as “t-coding” meaning that the sum of all n_i's is 0 modulo t, and n_i \equiv i (\mod~ t). Anyway, once the “hook expansion” is derived using Macdonald’s formula for the spaciel case of squares of odd integers it extends automatically to all \beta's. The marked hook formula is derived by looking at the coefficients of x^n \beta^{n-1}.

Update (July 17 ): Dominique Foata brought to my attention a special web page on these developments.