|
%I #19 May 15 2024 17:15:46
%S 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,
%T 13756,13833,346,1,0,1,1,1854,925705,6699824,748521,2252,1,0,1,1,
%U 14833,85394646,5691917785,3993445276,44127009,15184,1,0,1
%N Square array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.
%C 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).
%C T(n,k) is the maximal number of totally mixed Nash equilibria in games of k players, each with n+1 pure options.
%H Jeremy Tan, <a href="/A372307/b372307.txt">Antidiagonals n = 0..32, flattened</a>
%H 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.
%H 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.
%H B. H. Margolius, <a href="https://www.jstor.org/stable/3219303">The Dinner-Diner Matching Problem</a>, Mathematics Magazine, 76 (2003), pp. 107-118.
%H Jeremy Tan, <a href="https://gist.github.com/Parcly-Taxel/4c072993b7ae0aa9c165a5c779aef021">Python program</a>
%H 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.
%H 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).
%H <a href="/index/Ca#cardmatch">Index entries for sequences related to card matching</a>
%F 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).
%F T(n,k) ~ A089759(n,k)/exp(n).
%e Square array T(n,k) begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 0, 1, 2, 9, 44, ...
%e 1, 0, 1, 10, 297, 13756, ...
%e 1, 0, 1, 56, 13833, 6699824, ...
%e 1, 0, 1, 346, 748521, 3993445276, ...
%e 1, 0, 1, 2252, 44127009, 2671644472544, ...
%e 1, 0, 1, 15184, 2750141241, 1926172117389136, ...
%e 1, 0, 1, 104960, 178218782793, 1463447061709156064, ...
%t Table[Abs[Integrate[Exp[-x] LaguerreL[n, x]^(s-n), {x, 0, Infinity}]], {s, 0, 9}, {n, 0, s}] // Flatten
%o (Python) # See link.
%Y Rows 0-5 give A000012, A000166, A000459, A059073, A059074, A123297.
%Y Columns 0-4 give A000012, A000007, A000012, A000172, A371252.
%K nonn,tabl
%O 0,12
%A _Jeremy Tan_, Apr 26 2024
|
