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proposed

Jeremy Tan, <a href="/A372307/b372307_1.txt">Table of n, a(n) for Antidiagonals n = 0..56032, flattened</a>

B. H. Margolius, <a href="httphttps://www.jstor.org/stable/3219303">The Dinner-Diner Matching Problem</a>, Mathematics Magazine, 76 (2003), pp. 107-118.

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T(n,k) is the maximal number of totally mixed Nash equilibria in games of k players, each with n+1 pure options.

Raimundas Vidunas, <a href="https://arxiv.org/abs/1401.5400">MacMahon's master theorem and totally mixed Nash equilibria</a>, arXiv:1401.5400 [math.CO], 2014.

Raimundas Vidunas, <a href="https://doi.org/10.1007/s00026-017-0344-2">Counting derangements and Nash equilibria</a>, Ann. Comb. 21, No. 1, 131-152 (2017).

Jeremy Tan, <a href="https://gist.github.com/Parcly-Taxel/4c072993b7ae0aa9c165a5c779aef021">Python program</a>

(Python) See link.

Jeremy Tan, <a href="/A372307/b372307_1.txt">Table of n, a(n) for n = 0..560</a>

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1, 1, 1, 1, 1, 1, ... 1, 0, 1, 2, 9, 44, ...

1, 0, 1, 2, 9, 44, ...

T(n,k) ~ A089759(n,k)/exp(n).

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, ...

Table[Abs[Integrate[Exp[-x] LaguerreL[n, x]^(s-n), {x, 0, Infinity}]], {s, 0, 9}, {n, 0, s}] // Flatten

allocated for Jeremy TanSquare array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.

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

0,12

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).

Shalosh B. Ekhad, Christoph Koutschan and Doron Zeilberger, <a href="https://arxiv.org/abs/2101.10147">There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts]</a>, arXiv:2101.10147 [math.CO], 2021.

S. Even and J. Gillis, <a href="https://doi.org/10.1017/S0305004100052154">Derangements and Laguerre polynomials</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 79, Issue 1, January 1976, pp. 135-143.

B. H. Margolius, <a href="http://www.jstor.org/stable/3219303">The Dinner-Diner Matching Problem</a>, Mathematics Magazine, 76 (2003), pp. 107-118.

<a href="/index/Ca#cardmatch">Index entries for sequences related to card matching</a>

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).

Rows 0-5 give A000012, A000166, A000459, A059073, A059074, A123297.

Columns 0-4 give A000012, A000007, A000012, A000172, A371252.

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nonn,tabl

Jeremy Tan, Apr 26 2024

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