A proof of the Bender-Knuth conjecture
Let \(b_ r(n| I_ m)\) denote the number of r-rowed partitions of n whose parts lie in the set \(I_ m=\{1,2,...,m\}\) and decrease strictly along each row. \textit{E. A. Bender} and \textit{D. E. Knuth} conjectured [J. Comb. Theory, Ser. A 13, 40-54 (1972; Zbl 0246.05010)] that the generating function of \(b_ r(n| I_ m)\) is equal to \(\prod^{m}_{i=1}\prod^{i}_{j=1}(1-x^{r+i+j-1})/(1-x^{i+j- 1})\). This was derived from a related result by \textit{G. E. Andrews} [Pac. J. Math. 72, 283-291 (1977; Zbl 0376.10014)]. In this paper it is presented another, direct proof. It is based on the evaluation of certain determinants.
scientific article; zbMATH DE number 3845583