A multiset hook length formula and some applications (Q409408)

The first main theorem of the authors is the following multiset hook lenth formula: NEWLINETheorem 1. NEWLINELet \(t\) be a positive integer and \(\tau : \mathbb{Z} \to F\) be any weight function from \(\mathbb{Z}\) to a field \(F\). Then there exists a bijection NEWLINE\(\phi_t : \lambda \mapsto \vec{V}=(v_0,v_1,\ldots,v_{t-1})\) from NEWLINE\(t\)-cores onto \(V_t\)-codings such that NEWLINENEWLINE\[NEWLINE|\lambda| = \frac{1}{2t}(v_{0}^2 + v_{1}^2 +\dots+ v_{t-1}^2) - NEWLINE\frac{t^2 -1}{24} NEWLINE\]NEWLINE NEWLINEand NEWLINENEWLINE\[NEWLINE\prod_{h \in \mathcal{H}(\lambda)} \frac{\tau(h-t)\tau(h+t)}{\tau(h)^2} = \prod_{i=1}^{t-1} \frac{\tau(-i)^{\beta_i(\lambda)}} {\tau(i)^{\beta_i(\lambda)+t-i}} NEWLINE\prod_{0\leq i\leq j\leq t-1} \tau(v_j - v_i),NEWLINE\]NEWLINE NEWLINEwhere \(\beta_i (\lambda) = \# \{ \square \in \lambda:\;h(\square) = t-i \}\). NEWLINENEWLINENEWLINE NEWLINEThe special case \(\tau(x) = x\) gives a formula found by the second author in an earlier paper NEWLINE[\textit{G.-N. Han}, ``Some conjectures and open problems on partition hook lengths,'' NEWLINEExp. Math. 18, No.\,1, 97--106 (2009; Zbl 1167.05004); ``The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications,'' Ann. Inst. Fourier 60, No. 1, 1--29 (2010; Zbl 1215.05013)]. The special case \(\tau(x) = \sin(x)\) gives Theorem 2 which is the authors' second result in this paper. Among the consequences of the main results, the authors reproduce two formulas of Nekrasov-Okounkov and a formula of Iqbal. Moreover, Theorem 2 can be seen as a unified formula for the Nekrasov-Okounkov and Jacobi's triple product identity.