A059073 Card-matching numbers (Dinner-Diner matching numbers).

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

Offset: 0

Author

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

Keywords

• nonn
• nice

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 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).

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

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

Examples

			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.
		

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. 174-178.
• R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..100
• Shalosh B. Ekhad, Christoph Koutschan, and Doron Zeilberger, There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts], arXiv:2101.10147 [math.CO], 2021.
• F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
• Barbara H. Margolius, Dinner-Diner Matching Probabilities
• Barbara H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
• R. D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72 (1997), 411-425.
• S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.
• Raimundas Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arXiv preprint arXiv:1401.5400 [math.CO], 2014.
• Raimundas Vidunas, Counting derangements and Nash equilibria, Ann. Comb. 21, No. 1, 131-152 (2017).
• Index entries for sequences related to card matching

Crossrefs

Cf. A000166, A000459, A008290, A059056-A059071.

Programs

Maple

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);

Mathematica

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 *)

Formula

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.