On the generating functions of the Young lattice (Q1110514)

The Young lattice Y is the ordered set of all partitions with its order defined by the condition that, for two partitions \(\alpha =(\alpha_ 1,\alpha_ 2,...)\) and \(\beta =(\beta_ 1,\beta_ 2,...)\), \(\alpha\leq \beta\) means the inclusion relation of the corresponding Young diagrams; i.e., \(\alpha_ i\leq \beta_ i\) \((i=1,2,...)\). This is a ranked poset with its rank given by \(\alpha =\sum_{i\geq 0}\alpha_ i\). In this paper we give the generating function \[ f_{\lambda /\mu}(q):=\sum_{\mu \leq \xi \leq \lambda}q^{| \xi | -| \mu |} \] of the interval \([\mu,\lambda]:=\{\xi |\) \(\mu\leq \xi \leq \lambda \}\) as a determinant whose components involve Gaussian coefficients. The generating function for chains of the fixed length r in the interval is obtained similarly. These formulas were partly known to MacMahon, Hodge and Pedoe, Kreweras, and Carlitz. Geometrically, \(f_{\lambda /\mu}(q)\) is the Poincaré polynomial of a ``skew-Schubert variety''.