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A060540
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Square array read by antidiagonals downwards: T(n,k) = (n*k)!/(k!^n*n!), (n>=1, k>=1), the number of ways of dividing nk labeled items into n unlabeled boxes with k items in each box.
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29
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1, 1, 1, 1, 3, 1, 1, 10, 15, 1, 1, 35, 280, 105, 1, 1, 126, 5775, 15400, 945, 1, 1, 462, 126126, 2627625, 1401400, 10395, 1, 1, 1716, 2858856, 488864376, 2546168625, 190590400, 135135, 1, 1, 6435, 66512160, 96197645544, 5194672859376, 4509264634875, 36212176000, 2027025, 1
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OFFSET
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1,5
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COMMENTS
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The Copeland link gives the associations of this entry with the operator calculus of Appell Sheffer polynomials, the combinatorics of simple set partitions encoded in the Faa di Bruno formula for composition of analytic functions (formal Taylor series), the Pascal matrix, and the geometry of the n-dimensional simplices (hypertriangles, or hypertetrahedra). These, in turn, are related to simple instances of the application of the exponential formula / principle / schema giving the number of not-necessarily-connected objects composed from an ensemble of connected objects. - Tom Copeland, Jun 09 2021
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LINKS
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FORMULA
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T(n,k) = Product_{j=2..n} binomial(j*k-1,k-1). - M. F. Hasler, Aug 22 2014
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EXAMPLE
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Array begins:
1, 1, 1, 1, 1, 1, ...
1, 3, 10, 35, 126, 462, ...
1, 15, 280, 5775, 126126, 2858856, ...
1, 105, 15400, 2627625, 488864376, 96197645544, ...
1, 945, 1401400, 2546168625, 5194672859376, 11423951396577720, ...
...
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MATHEMATICA
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T[n_, k_] := (n*k)!/(k!^n*n!);
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PROG
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(PARI) { i=0; for (m=1, 20, for (n=1, m, k=m - n + 1; write("b060540.txt", i++, " ", (n*k)!/(k!^n*n!))); ) } \\ Harry J. Smith, Jul 06 2009
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CROSSREFS
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Cf. A000217, A000292, A000332, A000389, A000579, A000580, A007318, A036040, A099174, A133314, A132440, A135278 (associations in Copeland link).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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