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

1, 0, 1, 346, 748521, 3993445276, 45131501617225, 964363228180815366, 35780355973270898382001, 2158610844939711892526650456, 201028342764877992289387752167601, 27708893753238763155350683269145066450, 5459844285803153226360263675364357481841881

Offset

0,4

Comments

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).

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

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]

References

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.

R.D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72:411-425, 1997.

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

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.

B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.

Barbara H. Margolius, Dinner-Diner Matching Probabilities

R. Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arXiv preprint arXiv:1401.5400 [math.CO], 2014.

Vidunas, Raimundas Counting derangements and Nash equilibria Ann. Comb. 21, No. 1, 131-152 (2017).

Index entries for sequences related to card matching

Formula

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.

Example

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.

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, 4)/4!^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, 4]/4!^n, {n, 0, 18}] // Flatten (* Jean-François Alcover, Oct 21 2013, after Maple *)

Crossrefs

Cf. A008290, A059056-A059071.

Cf. A000166, A059056-A059071.

Sequence in context: A225934 A287700 A281163 * A273805 A142907 A052163

Adjacent sequences: A059071 A059072 A059073 * A059075 A059076 A059077

Keyword

nonn,nice

Author

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

Status

approved