OFFSET
0,2
COMMENTS
This is a triangle of card matching numbers. A deck has 3 kinds of cards, n of each kind. The deck is shuffled and dealt in to 3 hands each with n cards. 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..3n). The probability of exactly k matches is T(n,k)/((3n)!/n!^3).
Rows have lengths 1,4,7,10,...
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
D. Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
Barbara H. Margolius, Dinner-Diner Matching Probabilities
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 (3 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.
EXAMPLE
There are 27 ways of matching exactly 2 cards when there are 2 cards of each kind and 3 kinds of card so T(2,2)=27.
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 7 do seq(coeff(f(t, 3, n), t, m)/n!^3, m=0..3*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}]; Table[ Coefficient[ f[t, 3, n], t, m]/n!^3, {n, 0, 5}, {m, 0, 3*n}] // Flatten (* Jean-François Alcover, Mar 04 2013, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabf,nice
AUTHOR
Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
STATUS
approved