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A059062 Card-matching numbers (Dinner-Diner matching numbers). 3
1, 0, 0, 0, 0, 0, 1, 1, 0, 25, 0, 100, 0, 100, 0, 25, 0, 1, 2252, 15150, 48600, 99350, 144150, 156753, 131000, 87075, 45000, 19300, 6000, 1800, 250, 75, 0, 1, 44127009, 274314600, 822998550, 1583402400, 2189652825, 2311947008 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
COMMENTS
This is a triangle of card matching numbers. A deck has n kinds of cards, 5 of each kind. The deck is shuffled and dealt in to n hands with 5 cards each. 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..5n). The probability of exactly k matches is T(n,k)/((5n)!/(5!)^n).
Rows have lengths 1,6,11,16,...
Analogous to A008290 - Zerinvary Lajos, Jun 22 2005
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 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 j-th coefficient on x of the rook polynomial.
EXAMPLE
There are 25 ways of matching exactly 2 cards when there are 2 different kinds of cards, 5 of each so T(2,2)=25.
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 4 do seq(coeff(f(t, n, 5), t, m)/5!^n, m=0..5*n); od;
MATHEMATICA
nmax = 4; r[x_, n_, k_] := (k!^2*Sum[ x^j/((k-j)!^2*j!), {j, 0, 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, 5], t, m]/5!^n, {n, 0, nmax}, {m, 0, 5n}]](* Jean-François Alcover, Nov 23 2011, after Maple *)
CROSSREFS
Cf. A008290.
Sequence in context: A167624 A181614 A108321 * A179809 A110416 A028848
KEYWORD
nonn,tabf,nice
AUTHOR
Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
STATUS
approved

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Last modified May 28 15:12 EDT 2024. Contains 372916 sequences. (Running on oeis4.)