A059068 Card-matching numbers (Dinner-Diner matching numbers).

1, 9, 8, 6, 0, 1, 297, 672, 736, 480, 246, 64, 24, 0, 1, 13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1, 748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126

Offset

0,2

Comments

This is a triangle of card matching numbers. A deck has 4 kinds of cards, n of each kind. The deck is shuffled and dealt in to 4 hands with each with n cards. A match occurs for every card in the j-th hand of kind j. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..4n). The probability of exactly k matches is T(n,k)/((4n)!/n!^4).

Rows have lengths 1,5,9,13,...

Analogous to A008290 - Zerinvary Lajos, Jun 22 2005

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

Table of n, a(n) for n=0..36.

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

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

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

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 (4 in this case), k is the number of cards of each kind 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 736 ways of matching exactly 2 cards when there are 2 cards of each kind and 4 kinds of card so T(2,2)=736.
Triangle begins:
      1;
      9,     8,     6,     0,     1;
    297,   672,   736,   480,   246,    64,    24,    0,    1;
  13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1;
  ...

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);
for n from 0 to 5 do seq(coeff(f(t, 4, n), t, m)/n!^4, m=0..4*n); od;

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[ Coefficient[f[t, 4, n], t, m]/n!^4, {n, 0, 4}, {m, 0, 4*n}] // Flatten (* Jean-François Alcover, Dec 17 2012, translated from Maple *)

Crossrefs

Cf. A008290, A059056-A059071.

Cf. A008290.

Row sums give A008977.

Sequence in context: A200292 A155791 A327341 * A059069 A084660 A002391

Adjacent sequences: A059065 A059066 A059067 * A059069 A059070 A059071

Keyword

nonn,tabf,nice

Author

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

Status

approved