A059064 Card-matching numbers (Dinner-Diner matching numbers).

1, 1, 0, 1, 1, 0, 4, 0, 1, 1, 0, 9, 0, 9, 0, 1, 1, 0, 16, 0, 36, 0, 16, 0, 1, 1, 0, 25, 0, 100, 0, 100, 0, 25, 0, 1, 1, 0, 36, 0, 225, 0, 400, 0, 225, 0, 36, 0, 1, 1, 0, 49, 0, 441, 0, 1225, 0, 1225, 0, 441, 0, 49, 0, 1, 1, 0, 64, 0, 784, 0, 3136, 0, 4900, 0

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 2 kinds of cards, n of each kind. The deck is shuffled and dealt in to 2 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..2n). An odd number of matches is impossible, so alternating elements in each row of the triangle are zero. The probability of exactly k matches is T(n,k)/((2n)!/n!^2).

Rows have lengths 1,3,5,7,...

Analogous to A008290 - Zerinvary Lajos, Jun 22 2005

Examples

			There are 4 ways of matching exactly 2 cards when there are 2 cards of each kind and 2 kinds of card so T(2,2)=4.
		

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

• 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 10 do seq(coeff(f(t,2,n),t,m)/n!^2,m=0..2*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[ Table[ Coefficient[f[t, 2, n], t, m]/n!^2, {m, 0, 2n}], {n, 0, 8}] // Flatten  (* Jean-François Alcover, Jan 25 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 (2 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 x^j of the rook polynomial.