Determinants of super-Schur functions, lattice paths, and dotted plane partitions
This is an important contribution to the combinatorics of plane partitions, of nonintersecting lattice paths, and to combinatorial determinants. A large variety of results is obtained in a uniform way. The subject of this paper are minors of the matrix \[ {\mathcal S} ({\mathbf x}, {\mathbf y}) : =(s_{(k)} (x_ 1,\dots,x_ n/y_ 1,\dots,y_ n))_{n,k \in \mathbb{N}}, \] where \(s_{(k)} (x_ 1, \dots, x_ n/y_ 1, \dots, y_ n) = \sum_{i=0}^{k} h_ j(x_ 1, \dots,x_ n) e_{k-j} (y_ 1,\dots,y_ n)\) with \(h_ j\) and \(e_ j\) denoting the complete homogeneous and elementary symmetric functions, respectively. The \(s_{(k)}\) are special super-Schur functions (see e.g. \textit{V. G. Kac} [Adv. Math. 26, 8-96 (1977; Zbl 0366.17012)]). The focus is on giving interesting combinatorial interpretations for the minors of \({\mathcal S}({\mathbf x},{\mathbf y})\). The most straightforward interpretation is in terms of generating functions for nonintersecting lattice paths with three types of steps. As in the case of two types of steps (see \textit{I. M. Gessel} and \textit{X. G. Viennot}, ``Determinants, paths, and plane partitions'', preprint; or \textit{J. R. Stembridge} [Adv. Math. 83, No. 1, 96-131 (1990; Zbl 0790.05007)]), these families of nonintersecting lattice paths can be translated into certain plane partitions. The plane partitions under consideration in the paper under review are plane partitions whose entries are from two distinct copies of the set of integers. Since the integers from the second copy are written as dotted integers, the author calls his plane partitions dotted plane partitions. Subsequently, several specializations of \({\mathcal S}({\mathbf x},{\mathbf y})\) are considered. Interpretations in terms of generating functions for dotted plane partitions are obtained for many determinants of symmetric functions, of \(q\)-binomial coefficients, and of \(q\)-Stirling numbers of the first and second kind, respectively. Many of these interpretations are new. For the reviewer the most interesting are the interpretations for determinants of the \(q\)-Stirling numbers. These determinants are also given nice interpretations in terms of permutations and set partitions, respectively (which are derived from the plane partition interpretations). Thus, the author e.g. proves that minors of the matrices \((c_ q (n,k))_{n,k \in \mathbb{N}}\) and \((S_ q (n,k))_{n,k \in \mathbb{N}}\) of \(q\)-Stirling numbers of the first and second kind, respectively, are polynomials in \(q\) with nonnegative integer coefficients which had been conjectured previously (see e.g. \textit{B. E. Sagan} [Trans. Am. Math. Soc. 329, No. 2, 795-811 (1992)]).
scientific article; zbMATH DE number 221238