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A059064
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Card-matching numbers (Dinner-Diner matching numbers).
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0
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1, 1, 0, 1, 1, 0, 4, 0, 1, 1, 0, 9, 0, 9, 0, 1, 1, 0, 16, 0, 36, 0, 16, 0, 1, 1, 0, 25, 0, 100, 0, 100, 0, 25, 0, 1, 1, 0, 36, 0, 225, 0, 400, 0, 225, 0, 36, 0, 1, 1, 0, 49, 0, 441, 0, 1225, 0, 1225, 0, 441, 0, 49, 0, 1, 1, 0, 64, 0, 784, 0, 3136, 0, 4900, 0
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OFFSET
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0,7
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COMMENTS
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This is a triangle of card matching numbers. A deck has 2 kinds of cards, n of each kind. The deck is shuffled and dealt in to 2 hands each with n cards. 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). An odd number of matches is impossible, so alternating elements in each row of the triangle are zero. The probability of exactly k matches is T(n,k)/((2n)!/n!^2).
Rows have lengths 1,3,5,7,...
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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 (2 in this case), k is the number of cards of each kind 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 x^j of the rook polynomial.
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EXAMPLE
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There are 4 ways of matching exactly 2 cards when there are 2 cards of each kind and 2 kinds of card so T(2,2)=4.
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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 10 do seq(coeff(f(t, 2, n), t, m)/n!^2, m=0..2*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}]; Table[ Table[ Coefficient[f[t, 2, n], t, m]/n!^2, {m, 0, 2n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 25 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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