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A059057 Penrice Christmas gift numbers, Card-matching numbers (Dinner-Diner matching numbers). 0
1, 0, 0, 2, 4, 0, 16, 0, 4, 80, 192, 216, 128, 96, 0, 8, 4752, 10752, 11776, 7680, 3936, 1024, 384, 0, 16, 440192, 975360, 1035680, 696320, 329600, 114176, 31040, 5120, 1280, 0, 32, 59245120, 129054720, 135477504, 90798080 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
This is a triangle of card matching numbers. Two decks each have n kinds of cards, 2 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..2n). The probability of exactly k matches is T(n,k)/(2n)!.
Rows are of length 1,3,5,7,...
REFERENCES
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.
LINKS
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.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.
FORMULA
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 2) 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.
EXAMPLE
There are 16 ways of matching exactly 2 cards when there are 2 different kinds of cards, 2 of each in each of the two decks so T(2,2)=16.
MAPLE
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 6 do seq(coeff(f(t, n, 2), t, m), m=0..2*n); od;
MATHEMATICA
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}]; Flatten[ Table[ Coefficient[ f[t, n, 2], t, m], {n, 0, 6}, {m, 0, 2 n}]](* Jean-François Alcover, Nov 28 2011, translated from Maple *)
CROSSREFS
Sequence in context: A078022 A203850 A106603 * A196225 A127511 A321956
KEYWORD
nonn,tabf,nice
AUTHOR
Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)