0,4

A deck has n kinds of cards, 2 of each kind. The deck is shuffled and dealt in to n hands with 2 cards each. A match occurs for every card in the j-th hand of kind j. Then a(n) is the number of ways of achieving no matches. The probability of no matches is A(n)/((2n)!/2!^n).

Number of fixed-point-free permutations of n distinct letters (ABCD...), each of which appears twice. If there is only one letter of each type we get A000166. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 15 2006

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.

Barbara H. Margolius, Dinner-Diner Matching Probabilities

Index entries for sequences related to card matching

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 (2 in this case) 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 10 ways of achieving zero matches when there are 2 cards of each kind and 3 kinds of card so a(3)=10.

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);

seq(f(0, n, 2)/2!^n, n=0..18);

Essentially the same sequence as A000459.

Cf. A008290, A059056-A059071, A000166, A059073.

Sequence in context: A077281 A049387 A024295 this_sequence A000459 A125288 A069672

Adjacent sequences: A059069 A059070 A059071 this_sequence A059073 A059074 A059075

nonn

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