A combinatorial problem and the generalized cosh

Abstract

For j=1, ..., r, let h _j and k _j be integers such that 0≤h _j≤k_j−1 and ω _j=exp [2πi / k_j]. Then, the number of ways of placing n≥0 different balls into r distinct cells such that, for j=1, ..., r, the number of balls in the j ^th cell is congruent to h _j modulo k _j, is

$$\left( {\begin{array}{*{20}c}r \\{\Pi k_j } \\{j = 1} \\\end{array} } \right)^{ - 1} \mathop \Sigma \limits_{\begin{array}{*{20}c}{S_1 ,...,S_r } \\{0 \leqslant S_j \leqslant k_j - 1} \\\end{array} } \left( {\begin{array}{*{20}c}r \\{\Pi \omega _j } \\{j = 1} \\\end{array} - h_j s_j } \right)\left( {\begin{array}{*{20}c}r \\{\Sigma \omega _j } \\{j = 1} \\\end{array} s_j } \right)^n$$

. The proof is by means of the exponential enumerator and employs the generalized cosh: $C_k \left( x \right) = \sum\limits_{n = 0}^\infty {\frac{{x^{kn} }}{{\left( {kn} \right)!}}}$.