|
%I #40 Aug 28 2023 08:28:58
%S 1,0,1,56,13833,6699824,5691917785,7785547001784,16086070907249329,
%T 47799861987366600992,196500286135805946117201,
%U 1082973554682091552092493880,7797122311868240909226166565881
%N Card-matching numbers (Dinner-Diner matching numbers).
%C A deck has n kinds of cards, 3 of each kind. The deck is shuffled and dealt in to n hands with 3 cards each. A match occurs for every card in the j-th hand of kind j. A(n) is the number of ways of achieving no matches. The probability of no matches is A(n)/((3n)!/3!^n).
%C Number of fixed-point-free permutations of n distinct letters (ABCD...), each of which appears thrice. If there is only one letter of each type we get A000166. - _Zerinvary Lajos_, Oct 15 2006
%C a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 4 pure options. - Raimundas Vidunas, Jan 22 2014
%D F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
%D R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
%H Michael De Vlieger, <a href="/A059073/b059073.txt">Table of n, a(n) for n = 0..100</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 F. F. Knudsen and I. Skau, <a href="http://www.jstor.org/stable/2691467">On the Asymptotic Solution of a Card-Matching Problem</a>, Mathematics Magazine 69 (1996), 190-197.
%H Barbara H. Margolius, <a href="http://academic.csuohio.edu/bmargolius/homepage/dinner/dinner/cardentry.htm">Dinner-Diner Matching Probabilities</a>
%H Barbara H. Margolius, <a href="http://www.jstor.org/stable/3219303">The Dinner-Diner Matching Problem</a>, Mathematics Magazine, 76 (2003), 107-118.
%H R. D. McKelvey and A. McLennan, <a href="https://doi.org/10.1006/jeth.1996.2214">The maximal number of regular totally mixed Nash equilibria</a>, J. Economic Theory, 72 (1997), 411-425.
%H S. G. Penrice, <a href="http://www.jstor.org/stable/2324927">Derangements, permanents and Christmas presents</a>, The American Mathematical Monthly 98(1991), 617-620.
%H Raimundas Vidunas, <a href="https://arxiv.org/abs/1401.5400">MacMahon's master theorem and totally mixed Nash equilibria</a>, arXiv preprint 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 G.f.: Sum_{j=0..n*k} coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)! where n is the number of kinds of cards, k is the number of cards of each kind (3 in this case) and R(x, n, k) is the rook polynomial given by R(x, n, k) = k!^2*Sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.
%e There are 56 ways of achieving zero matches when there are 3 cards of each kind and 3 kinds of card so a(3)=56.
%p p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k); seq(f(0,n,3)/3!^n,n=0..18);
%t p[x_, k_] := k!^2*Sum[x^j/((k-j)!^2*j!), {j, 0, k}]; R[x_, n_, k_] := p[x, k]^n; f[t_, n_, k_] := Sum[Coefficient[R[x, n, k], x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Table[f[0, n, 3]/3!^n, {n, 0, 12}] (* _Jean-François Alcover_, May 21 2012, translated from Maple *)
%Y Cf. A000166, A000459, A008290, A059056-A059071.
%K nonn,nice
%O 0,4
%A Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
|
