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A combinatorial problem and the generalized cosh

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1036)

Abstract

For j=1, ..., r, let h j and k j be integers such that 0≤h j≤kj−1 and ω j=exp [2πi / kj]. 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: \(C_k \left( x \right) = \sum\limits_{n = 0}^\infty {\frac{{x^{kn} }}{{\left( {kn} \right)!}}}\).

Keywords

  • Formal Power Series
  • Sine Function
  • Distinct Cell
  • Chinese Remainder Theorem
  • Exponential Generate

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1983 Springer-Verlag

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Cornish, W.H. (1983). A combinatorial problem and the generalized cosh. In: Casse, L.R.A. (eds) Combinatorial Mathematics X. Lecture Notes in Mathematics, vol 1036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071516

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  • DOI: https://doi.org/10.1007/BFb0071516

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