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A059069 Card-matching numbers (Dinner-Diner matching numbers). 0
1, 9, 8, 6, 0, 1, 4752, 10752, 11776, 7680, 3936, 1024, 384, 0, 16, 17927568, 64105344, 109524960, 117863424, 89474544, 49828608, 21352896, 6718464, 1854576, 279936, 69984, 0, 1296, 248341303296, 1215287525376 (list; graph; refs; listen; history; text; internal format)
This is a triangle of card matching numbers. Two decks each have 4 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..4n). The probability of exactly k matches is T(n,k)/(4n)!.
rows are of length 1,5,9,13,...
F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
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, pp. 174-178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
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.
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 (4 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.
There are 11776 ways of matching exactly 2 cards when there are 2 cards of each kind and 4 kinds of card so T(2,2)=11776.
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, 4, n), t, m), m=0..4*n); od;
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, 4, n], t, m], {n, 0, 5}, {m, 0, 4*n}] // Flatten (* Jean-François Alcover, Oct 21 2013, after Maple *)
Sequence in context: A155791 A327341 A059068 * A084660 A002391 A193626
Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

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Last modified June 10 08:15 EDT 2023. Contains 363196 sequences. (Running on oeis4.)