Partitions with prescribed hook differences
The hook difference at the (i,j)-th node of Ferrers graph of a partition P is defined as the number of nodes in the i-th row of P minus the number of nodes in the j-th column of P. The (i,j)-th node lies on diagonal c if \(i-j=c\). Let \(p_{k,i}(N,M;\alpha,\beta;n)\) be the number of partitions of n into at most M parts each \(\leq\) N that the hook difference on diagonal 1-\(\beta\) are \(\geq -i+\beta +1\) and on diagonal \(\alpha\)-1 are \(\leq K-i-\alpha -1.\) Theorem 1. Let \(1\leq i\leq K/2,\alpha >0\), \(\beta >0\), \(\alpha +\beta <K\) with M,N,i,\(\alpha\),\(\beta\) integers. For \(-i+\beta \leq N-M\leq K-i- \alpha,\) \[ \sum_{n\geq 0}P_{K,i}(N,M;\alpha,\beta;n)q^ n=\sum^{\infty}_{\mu =-\infty}q^{\mu (K\mu +i)(\alpha +\beta)- K\beta \mu}\left[ \begin{matrix} N+M\\ N-K\mu \end{matrix} \right]_ q- \sum^{\infty}_{\mu =-\infty}q^{\mu (k\mu -i)(\alpha +\beta)-K\beta \mu +\beta i}\left[ \begin{matrix} N+M\\ N-K\mu +i\end{matrix} \right]_ q \] where \(\left[\begin{matrix} A\\ B\end{matrix} \right]_ q\) is the Gaussian q-binomial coefficient. Theorem 2. For \(1\leq i\leq K/2,\alpha >0\), \(\beta >0\), \(\alpha +\beta <K\) all integers \[ \sum^{\infty}_{n=0}P_{K,i}(\alpha,\beta;n)q^ n=\frac{1}{(q)_{\infty}}\sum^{\infty}_{\mu =-\infty}q^{\mu (K\mu -i)(\alpha +\beta)+K\beta \mu}(1-q^{\beta i+2i(\alpha +\beta)\mu)}= \] \[ \frac{1}{(q)_{\infty}}(\prod^{\infty}_{n=0}(1-q^{2K(\alpha +\beta)(n+1)})(1+q^{K\beta +(\alpha +\beta)(2Kn+K-i)})(1+q^{-K\beta +(\alpha +\beta)(2Kn+K+i)}- \] \[ q^{\beta i}\prod^{\infty}_{n=0}(1-q^{2K(\alpha +\beta)(n+1)})(1+q^{K\beta +(\alpha +\beta)(2Kn+K+i)})(1+q^{-K\beta +(\alpha +\beta)(2Kn+K-i)}) \] where \((q)_{\infty}=\prod^{\infty}_{n=1}(I-q^ n)\), \(P_{K,i}(\alpha,\beta;n)=\lim_{M,N\to \infty}P_{K,i}(N,M;\alpha,\beta;n).\) Theorem 3. For \(1\leq i\), \(\alpha\leq K/2\), 2i\(\neq K\), \[ \sum^{\infty}_{n=0}P_{K,i}(\alpha,\alpha;n)q^ n=\prod^{\infty}_{n=1,n\not\equiv 0,\pm \alpha_ i(mod \alpha K)}(1- q^ n)^{-1}; \] Theorem 4. Let K,\(\alpha\),\(\beta\) be positive integers with \(\alpha +\beta <3K\). Let \(A_ k\) (\(\alpha\),\(\beta\),n) denote the number of partitions of n into parts not congruent with \(0,\pm K\beta (mod 2K(\alpha +\beta))\) nor to \(\pm 2K^{\beta}(mod 4K(\alpha +\beta))\). Then \(P_{3K,K}\quad (\alpha,\beta;n)=A_ K(\alpha,\beta,n).\) Theorem 5. Let K,\(\alpha\),\(\beta\) be positive integers with \(\alpha +\beta <2K\). Let \(B_ k(\alpha,\beta,n)\) denote the number of partitions of n into parts not congruent to \(0,\pm K\beta mod (K(\alpha +\beta))\). Then \(P_{2K,K}(\alpha,\beta;n)=B_ K(\alpha,\beta,n).\)
scientific article; zbMATH DE number 4047728