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
. 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
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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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