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A059074 Card-matching numbers (Dinner-Diner matching numbers). 0


%S 1,0,1,346,748521,3993445276,45131501617225,964363228180815366,

%T 35780355973270898382001,2158610844939711892526650456,

%U 201028342764877992289387752167601,27708893753238763155350683269145066450,5459844285803153226360263675364357481841881

%N Card-matching numbers (Dinner-Diner matching numbers).

%C A deck has n kinds of cards, 4 of each kind. The deck is shuffled and dealt in to n hands with 4 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)/((4n)!/4!^n).

%C Number of fixed-point-free permutations of n distinct letters (ABCD...), each of which appears 4 times: 1111, 11112222, 111122223333, 1111222233334444, etc. If there is only one letter of each type we get A000166 - _Zerinvary Lajos_, Nov 05 2006

%C a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 5 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 R.D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72:411-425, 1997.

%D S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.

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

%H Barbara H. Margolius, <a href="http://academic.csuohio.edu/bmargolius/homepage/dinner/dinner/cardentry.htm">Dinner-Diner Matching Probabilities</a>

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

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

%F G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) 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 346 ways of achieving zero matches when there are 4 cards of each kind and 3 kinds of card so A(3)=346.

%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,4)/4!^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, 4]/4!^n, {n, 0, 18}] // Flatten (* _Jean-Fran├žois Alcover_, Oct 21 2013, after Maple *)

%Y Cf. A008290, A059056-A059071.

%Y Cf. A000166, A059056-A059071.

%K nonn,nice

%O 0,4

%A Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

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Last modified August 11 00:32 EDT 2020. Contains 336403 sequences. (Running on oeis4.)