0,7

This is a triangle of card matching numbers. 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. 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)!/2^n).

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

Analogous to A008290. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 10 2005

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, 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.

There are 4 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)=4.

Triangle begins:

1

"0", 0, 1

1, '0', "4", 0, 1

10, 24, 27, '16', "12", 0, 1

297, 672, 736, 480, 246, '64', "24", 0, 1

13756, 30480, 32365, 21760, 10300, 3568, 970, '160', "40", 0, 1

925705, 2016480, 2116836, 1418720, 677655, 243360, 67920, 14688, 2655, '320', "60", 0, 1

No. 1. diagonal : " " :0, 4, 12, 24, 40, 60, 84, 112, 144, 180, equals A046092

No. 2. diagonal : ' ' :0, 16, 64, 160, 320, 560, 896, 1344, equals A102860

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, n, 2), t, m)/2^n, m=0..2*n); od;

Cf. A059056-A059071, A008290.

Sequence in context: A079642 A121408 A121301 this_sequence A127153 A054375 A136452

Adjacent sequences: A059053 A059054 A059055 this_sequence A059057 A059058 A059059

nonn,tabf,nice

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

Additional comments from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 18 2007