A proof of the Bender-Knuth conjecture
Basil Gordon · Pacific Journal of Mathematics · 1983
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,... ,/w} and decrease strictly along each row.It is shown that mi " = n no-* r+i+ '-')/o -n=0 100 BASIL GORDON 2. Notation.If n is a positive integer, while a and y are indeterminates, we write (a; y) n = (1 -a){\ -ay){\ -ay 2 ) • • • (l -ay"-*).By convention, (a; y) 0 = l.IfO X 2 >:--->:X r >:0.We consider r-rowed partitions of the type enumerated by b r (n \ 5), but where there are exactly X t non-zero parts in the ith row (/ = 1,... ,r).Let b r (n; \ l9 ... ,X r | 5) be the number of such partitions of «, and put B,(x;\ l9 ... 9 \ r \S)= 1 ^(/ijX!,...,^!^".Clearly B r (x\ S) = S^j^x; X,,...,X r | S), where the sum is extended over all sequences (X,.) with X, > • • • > X r > 0. We now obtain an expression for B r (x; X 1? ... ,X r | I m ) as a determinant.THEOREM 1.