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A059061
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Card-matching numbers (Dinner-Diner matching numbers).
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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
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OFFSET
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0,6
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COMMENTS
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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
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F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
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LINKS
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FORMULA
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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
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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
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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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MATHEMATICA
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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[ Coefficient[ r[x, n, k], x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; a[n_, m_] := Coefficient[ f[t, n, 4], t, m]; Table[a[n, m], {n, 0, 4}, {m, 0, 4*n}] // Flatten (* Jean-François Alcover, Oct 07 2013, translated from Maple *)
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CROSSREFS
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
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STATUS
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approved
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