Coefficients associated with the expansion of certain products

Abstract

Simple combinatorial interpretations are given for the coefficients of the polynomials H n ( x , y ) {H_n}(x,y) and G n ( x , y ) {G_n}(x,y) defined by ∏ ( 1 + x n y m t ) = Σ G n ( x , y ) t n / ( x ) n ( y ) n \prod (1 + {x^n}{y^m}t) = \Sigma {G_n}(x,y){t^n}/{(x)_n}{(y)_n} and ∏ ( 1 − x n y m t ) − 1 = Σ H n ( x , y ) t n / ( x ) n ( y ) n \prod {(1 - {x^n}{y^m}t)^{ - 1}} = \Sigma {H_n}(x,y){t^n}/{(x)_n}{(y)_n} .