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A059061
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Card-matching numbers (Dinner-Diner matching numbers).
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0
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1, 0, 0, 0, 0, 24, 576, 0, 9216, 0, 20736, 0, 9216, 0, 576, 4783104, 25214976, 62705664, 98648064, 109859328, 87588864, 54411264, 23887872, 9455616, 1769472, 663552, 0, 13824, 248341303296, 1215287525376, 2855842873344
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| This is a triangle of card matching numbers. Two decks each have n kinds of cards, 4 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..4n). The probability of exactly k matches is T(n,k)/(4n)!.
rows are of length 1,5,9,13,...
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REFERENCES
| 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.
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LINKS
| Index entries for sequences related to card matching
Barbara H. Margolius, Dinner-Diner Matching Probabilities
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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 4) 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.
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EXAMPLE
| There are 9216 ways of matching exactly 2 cards when there are 2 different kinds of cards, 4 of each in each of the two decks so T(2,2)=9216.
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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 4 do seq(coeff(f(t, n, 4), t, m), m=0..4*n); od;
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CROSSREFS
| Cf. A008290, A059056-A059071.
Sequence in context: A158538 A171329 A077423 * A206991 A206933 A206868
Adjacent sequences: A059058 A059059 A059060 * A059062 A059063 A059064
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KEYWORD
| nonn,tabf,nice
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AUTHOR
| Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
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