A059058 Card-matching numbers (Dinner-Diner matching numbers).

1, 0, 0, 0, 1, 1, 0, 9, 0, 9, 0, 1, 56, 216, 378, 435, 324, 189, 54, 27, 0, 1, 13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1, 6699824, 23123880, 38358540, 40563765, 30573900, 17399178, 7723640

Offset

0,8

Comments

This is a triangle of card matching numbers. 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. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..3n). The probability of exactly k matches is T(n,k)/((3n)!/(3!)^n).

Rows have lengths 1,4,7,10,...

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

Vincenzo Librandi, Rows n = 1..30, flattened

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, k is the number of cards of each kind (here k is 3) 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 j-th coefficient on x of the rook polynomial.

Example

There are 9 ways of matching exactly 2 cards when there are 2 different kinds of cards, 3 of each in each of the two decks so T(2,2)=9.

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 6 do seq(coeff(f(t, n, 3), t, m)/3!^n, m=0..3*n); od;

Mathematica

p[x_, k_] := k!^2*Sum[ x^j/((k-j)!^2*j!), {j, 0, k}]; f[t_, n_, k_] := Sum[ Coefficient[ p[x, k]^n, x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Flatten[ Table[ Coefficient[ f[t, n, 3], t, m]/3!^n, {n, 0, 6}, {m, 0, 3n}]] (* Jean-François Alcover, Jan 31 2012, after Maple *)

Crossrefs

Cf. A008290, A059056-A059071, A008290.

Sequence in context: A086199 A167545 A272965 * A343587 A021015 A010680

Adjacent sequences: A059055 A059056 A059057 * A059059 A059060 A059061

Keyword

nonn,tabf,nice

Author

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

Status

approved