A059057 Penrice Christmas gift numbers, Card-matching numbers (Dinner-Diner matching numbers).

1, 0, 0, 2, 4, 0, 16, 0, 4, 80, 192, 216, 128, 96, 0, 8, 4752, 10752, 11776, 7680, 3936, 1024, 384, 0, 16, 440192, 975360, 1035680, 696320, 329600, 114176, 31040, 5120, 1280, 0, 32, 59245120, 129054720, 135477504, 90798080

Offset: 0

Author

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

Keywords

• nonn
• tabf
• nice

Comments

This is a triangle of card matching numbers. Two decks each have n kinds of cards, 2 of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..2n). The probability of exactly k matches is T(n,k)/(2n)!.

Rows are of length 1,3,5,7,...

Examples

			There are 16 ways of matching exactly 2 cards when there are 2 different kinds of cards, 2 of each in each of the two decks so T(2,2)=16.
		

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.

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 6 do seq(coeff(f(t,n,2),t,m),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}]; Flatten[ Table[ Coefficient[ f[t, n, 2], t, m], {n, 0, 6}, {m, 0, 2 n}]](* Jean-François Alcover, Nov 28 2011, 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, k is the number of cards of each kind (here k is 2) 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.