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A059063
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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, 0, 120, 14400, 0, 360000, 0, 1440000, 0, 1440000, 0, 360000, 0, 14400, 3891456000, 26179200000, 83980800000, 171676800000, 249091200000, 270869184000, 226368000000, 150465600000, 77760000000
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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. Two decks each have n kinds of cards, 5 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)!.
Rows are of length 1,6,11,16,...
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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 5) 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 360000 ways of matching exactly 2 cards when there are 2 different kinds of cards, 5 of each in each of the two decks so T(2,2)=360000.
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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, 5), 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}]; k = 5; Table[ Table[ Coefficient[f[t, n, k], t, m], {m, 0, k*n}], {n, 0, 4}] // Flatten (* Jean-François Alcover, Oct 21 2013, after 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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