0,2

This is a triangle of card matching numbers. Two decks each have 3 kinds of cards, n 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..3n). The probability of exactly k matches is T(n,k)/(3n)!.

rows are of length 1,4,7,10,...

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.

F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.

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.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 174-178.

R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

Index entries for sequences related to card matching

Barbara H. Margolius, Dinner-Diner Matching Probabilities

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.

There are 216 ways of matching exactly 2 cards when there are 2 cards of each kind and 3 kinds of card so T(2,2)=216.

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, 3, n), t, m), m=0..3*n); od;

Cf. A008290, A059056-A059071.

Sequence in context: A004443 A008290 A059066 this_sequence A065861 A126832 A068908

Adjacent sequences: A059064 A059065 A059066 this_sequence A059068 A059069 A059070

nonn,tabf,nice

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