On the difference of successive Gaussian polynomials
Let \(M,N\in\mathbb{Z}\); as usual, define the ``Gaussian polynomials'' by \({N\brack M}=\prod^ M_ 1(1-q^{N+1-j})\) \((1-q^ j)^{-1}\), if \(0\leq M\leq N\). Otherwise, put \({N\brack M}=0\). According to the author's introduction, ``the object in this paper is to relate differences of Gaussian polynomials to partitions through the use of Frobenius symbols''. [See Section 2 of the author's work Generalized Frobenius partitions (Mem. Am. Math. Soc. 301, 1984; Zbl 0544.10010)]. The Frobenius symbol of any partition \(\pi\) (of a positive integer) ``is constructed as follows: In the Ferrers graph of \(\pi\) delete the main diagonal (of say \(r\) nodes); then create a two-line array of integers wherein the upper row consists of the cardinalities of the \(r\) rows in the Ferrers graph to the right of the main diagonal and the lower row consists of the cardinalities of the \(r\) columns in the Ferrers graph below the main diagonal. For example, if \(\pi\) is the partition 5+4+4+2, then the ... Frobenius symbol is \({4 2 1\choose 3 2 0}\).'' The author proves the theorem: For \(0\leq j\leq N\), \(q^{- j}\bigl({N+j\brack j}-{N+j\brack j-1}\bigr)\) is the generating function for partitions \(\pi\) with Frobenius symbols \({a_ 1\cdots a_ r\choose b_ 1\cdots b_ r}\) satisfying \(a_ 1<N-1\), \(b_ 1<j-1\) and \(a_ i- b_ i<N-j\), for \(1\leq i\leq r\). Setting \(j=N\) in the theorem yields an interpretation of the \(q\)-Catalan numbers, \((1-q)(1-q^{N+1})^{- 1}{2N\brack N}\).
scientific article; zbMATH DE number 168224
scientific article