

A059058


Cardmatching numbers (DinnerDiner matching numbers).


7



1, 0, 0, 0, 1, 1, 0, 9, 0, 9, 0, 1, 56, 216, 378, 435, 324, 189, 54, 27, 0, 1, 13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1, 6699824, 23123880, 38358540, 40563765, 30573900, 17399178, 7723640
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OFFSET

0,8


COMMENTS

This is a triangle of card matching numbers. A deck has n kinds of cards, 3 of each kind. The deck is shuffled and dealt in to n hands with 3 cards each. A match occurs for every card in the jth hand of kind j. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..3n). The probability of exactly k matches is T(n,k)/((3n)!/(3!)^n).
Rows have lengths 1,4,7,10,...


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. 174178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.


LINKS



FORMULA

G.f.: sum(coeff(R(x, n, k), x, j)*(t1)^j*(n*kj)!, 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 3) and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((kj)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the jth coefficient on x of the rook polynomial.


EXAMPLE

There are 9 ways of matching exactly 2 cards when there are 2 different kinds of cards, 3 of each in each of the two decks so T(2,2)=9.


MAPLE

p := (x, k)>k!^2*sum(x^j/((kj)!^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)*(t1)^j*(n*kj)!, j=0..n*k);
for n from 0 to 6 do seq(coeff(f(t, n, 3), t, m)/3!^n, m=0..3*n); od;


MATHEMATICA

p[x_, k_] := k!^2*Sum[ x^j/((kj)!^2*j!), {j, 0, k}]; f[t_, n_, k_] := Sum[ Coefficient[ p[x, k]^n, x, j]*(t1)^j*(n*kj)!, {j, 0, n*k}]; Flatten[ Table[ Coefficient[ f[t, n, 3], t, m]/3!^n, {n, 0, 6}, {m, 0, 3n}]] (* JeanFrançois Alcover, Jan 31 2012, after Maple *)


CROSSREFS



KEYWORD

nonn,tabf,nice


AUTHOR

Barbara Haas Margolius (margolius(AT)math.csuohio.edu)


STATUS

approved



