

A059073


Cardmatching numbers (DinnerDiner matching numbers).


8



1, 0, 1, 56, 13833, 6699824, 5691917785, 7785547001784, 16086070907249329, 47799861987366600992, 196500286135805946117201, 1082973554682091552092493880, 7797122311868240909226166565881
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OFFSET

0,4


COMMENTS

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 jth 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).
Number of fixedpointfree 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
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]


REFERENCES

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.


LINKS

Table of n, a(n) for n=0..12.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a CardMatching Problem, Mathematics Magazine 69 (1996), 190197.
Barbara H. Margolius, DinnerDiner Matching Probabilities
B. H. Margolius, The DinnerDiner Matching Problem, Mathematics Magazine, 76 (2003), 107118.
R. D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72 (1997), 411425.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617620.
R. Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arXiv preprint arXiv:1401.5400 [math.CO], 2014.
Index entries for sequences related to card matching


FORMULA

G.f.: sum(coeff(R(x, n, k), x, j)*(t1)^j*(n*kj)!, 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/((kj)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.


EXAMPLE

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.


MAPLE

p := (x, k)>k!^2*sum(x^j/((kj)!^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)*(t1)^j*(n*kj)!, j=0..n*k); seq(f(0, n, 3)/3!^n, n=0..18);


MATHEMATICA

p[x_, k_] := k!^2*Sum[x^j/((kj)!^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]*(t1)^j*(n*kj)!, {j, 0, n*k}]; Table[f[0, n, 3]/3!^n, {n, 0, 12}] (* JeanFrançois Alcover, May 21 2012, translated from Maple *)


CROSSREFS

Cf. A000166, A000459, A008290, A059056A059071.
Sequence in context: A059989 A184125 A213865 * A246632 A289204 A201726
Adjacent sequences: A059070 A059071 A059072 * A059074 A059075 A059076


KEYWORD

nonn,nice


AUTHOR

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


STATUS

approved



