A059066 Card-matching numbers (Dinner-Diner matching numbers).

1, 2, 3, 0, 1, 10, 24, 27, 16, 12, 0, 1, 56, 216, 378, 435, 324, 189, 54, 27, 0, 1, 346, 1824, 4536, 7136, 7947, 6336, 3936, 1728, 684, 128, 48, 0, 1, 2252, 15150, 48600, 99350, 144150, 156753, 131000, 87075, 45000, 19300, 6000

Offset: 0

Author

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

Keywords

• nonn
• tabf
• nice

Comments

This is a triangle of card matching numbers. A deck has 3 kinds of cards, n of each kind. The deck is shuffled and dealt in to 3 hands 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..3n). The probability of exactly k matches is T(n,k)/((3n)!/n!^3).

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

Analogous to A008290. - Zerinvary Lajos, Jun 22 2005

Examples

			There are 27 ways of matching exactly 2 cards when there are 2 cards of each kind and 3 kinds of card so T(2,2)=27.
		

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

• D. Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4.
• 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

Crossrefs

Cf. A008290, A059056-A059071.

Cf. A008290.

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);
for n from 0 to 7 do seq(coeff(f(t,3,n),t,m)/n!^3,m=0..3*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, 3, n], t, m]/n!^3, {n, 0, 5}, {m, 0, 3*n}] // Flatten (* Jean-François Alcover, Mar 04 2013, translated from Maple *)

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 (3 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.