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A372307
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Square array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.
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1
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1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 9, 10, 1, 0, 1, 1, 44, 297, 56, 1, 0, 1, 1, 265, 13756, 13833, 346, 1, 0, 1, 1, 1854, 925705, 6699824, 748521, 2252, 1, 0, 1, 1, 14833, 85394646, 5691917785, 3993445276, 44127009, 15184, 1, 0, 1
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OFFSET
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0,12
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COMMENTS
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A deck has k suits of n cards each. The deck is shuffled and dealt into k hands of n cards each. A match occurs for every card in the i-th hand of suit i. T(n,k) is the number of ways of achieving no matches. The probability of no matches is T(n,k)/((n*k)!/n!^k).
T(n,k) is the maximal number of totally mixed Nash equilibria in games of k players, each with n+1 pure options.
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LINKS
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FORMULA
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T(n,k) = (-1)^(n*k) * Integral_{x=0..oo} exp(-x)*L_n(x)^k dx, where L_n(x) is the Laguerre polynomial of degree n (Even and Gillis).
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EXAMPLE
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Square array T(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 0, 1, 2, 9, 44, ...
1, 0, 1, 10, 297, 13756, ...
1, 0, 1, 56, 13833, 6699824, ...
1, 0, 1, 346, 748521, 3993445276, ...
1, 0, 1, 2252, 44127009, 2671644472544, ...
1, 0, 1, 15184, 2750141241, 1926172117389136, ...
1, 0, 1, 104960, 178218782793, 1463447061709156064, ...
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MATHEMATICA
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Table[Abs[Integrate[Exp[-x] LaguerreL[n, x]^(s-n), {x, 0, Infinity}]], {s, 0, 9}, {n, 0, s}] // Flatten
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PROG
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(Python) # See link.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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