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A059071
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Card-matching numbers (Dinner-Diner matching numbers) for 5 kinds of cards.
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23
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1, 44, 45, 20, 10, 0, 1, 440192, 975360, 1035680, 696320, 329600, 114176, 31040, 5120, 1280, 0, 32, 52097831424, 179811290880, 298276007040, 315423836640, 237742646400, 135296008128, 60059024640
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OFFSET
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0,2
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COMMENTS
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This is a triangle of card matching numbers. Two decks each have 5 kinds of cards, n 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..5n). The probability of exactly k matches is T(n,k)/(5n)!.
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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 (5 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 for x^j of the rook polynomial.
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EXAMPLE
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There are 1,035,680 ways of matching exactly 2 cards when there are 2 cards of each kind and 5 kinds of card so T(2,2)=1,035,680.
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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 3 do seq(coeff(f(t, 5, n), t, m), m=0..5*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[ Coefficient[ f[t, 5, n], t, m], {n, 0, 3}, {m, 0, 5*n}] // Flatten (* Jean-François Alcover, Mar 04 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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EXTENSIONS
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STATUS
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approved
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