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A059060
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Card-matching numbers (Dinner-Diner matching numbers).
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4
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1, 0, 0, 0, 0, 1, 1, 0, 16, 0, 36, 0, 16, 0, 1, 346, 1824, 4536, 7136, 7947, 6336, 3936, 1728, 684, 128, 48, 0, 1, 748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126, 453888
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| This is a triangle of card matching numbers. A deck has n kinds of cards, 4 of each kind. The deck is shuffled and dealt in to n hands with 4 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..4n). The probability of exactly k matches is T(n,k)/((4n)!/(4!)^n).
Rows have lengths 1,5,9,13,...
Analogous to A008290 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2005
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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
| Barbara H. Margolius, Dinner-Diner Matching Probabilities
Index entries for sequences related to card matching
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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 16 ways of matching exactly 2 cards when there are 2 different kinds of cards, 4 of each so T(2,2)=16.
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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 5 do seq(coeff(f(t, n, 4), t, m)/4!^n, m=0..4*n); od;
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CROSSREFS
| Cf. A008290, A059056-A059071.
Cf. A008290.
Sequence in context: A007791 A070570 A005077 * A059681 A135925 A188784
Adjacent sequences: A059057 A059058 A059059 * A059061 A059062 A059063
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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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