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A proof of the Bender-Knuth conjecture.(English)Zbl 0533.05005

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.E. A. Bender andD. 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 byG. 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.
Reviewer: A.Pethö

MSC:

05A15 Exact enumeration problems, generating functions
05A17 Combinatorial aspects of partitions of integers
11P81 Elementary theory of partitions

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