

A059068


Cardmatching numbers (DinnerDiner matching numbers).


0



1, 9, 8, 6, 0, 1, 297, 672, 736, 480, 246, 64, 24, 0, 1, 13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1, 748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126
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OFFSET

0,2


COMMENTS

This is a triangle of card matching numbers. A deck has 4 kinds of cards, n of each kind. The deck is shuffled and dealt in to 4 hands with each with n cards. 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..4n). The probability of exactly k matches is T(n,k)/((4n)!/n!^4).
Rows have lengths 1,5,9,13,...


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 (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/((kj)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.


EXAMPLE

There are 736 ways of matching exactly 2 cards when there are 2 cards of each kind and 4 kinds of card so T(2,2)=736.


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 5 do seq(coeff(f(t, 4, n), t, m)/n!^4, m=0..4*n); od;


MATHEMATICA

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[ Coefficient[r[x, n, k], x, j]*(t1)^j*(n*kj)!, {j, 0, n*k}]; Table[ Coefficient[f[t, 4, n], t, m]/n!^4, {n, 0, 4}, {m, 0, 4*n}] // Flatten (* JeanFrançois Alcover, Dec 17 2012, translated from Maple *)


CROSSREFS



KEYWORD

nonn,tabf,nice


AUTHOR

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


STATUS

approved



