%I #17 Sep 21 2017 03:36:05
%S 1,44,45,20,10,0,1,13756,30480,32365,21760,10300,3568,970,160,40,0,1,
%T 6699824,23123880,38358540,40563765,30573900,17399178,7723640,2729295,
%U 776520,180100,33372,5355,540,90,0,1
%N Card-matching numbers (Dinner-Diner matching numbers).
%C This is a triangle of card matching numbers. A deck has 5 kinds of cards, n of each kind. The deck is shuffled and dealt in to 5 hands with 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..5n). The probability of exactly k matches is T(n,k)/((5n)!/n!^5).
%C Rows are of length 1,6,11,16,...
%D F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
%D S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
%D R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
%H F. F. Knudsen and I. Skau, <a href="http://www.jstor.org/stable/2691467">On the Asymptotic Solution of a Card-Matching Problem</a>, Mathematics Magazine 69 (1996), 190-197.
%H B. H. Margolius, <a href="http://www.jstor.org/stable/3219303">The Dinner-Diner Matching Problem</a>, Mathematics Magazine, 76 (2003), 107-118.
%H Barbara H. Margolius, <a href="http://academic.csuohio.edu/bmargolius/homepage/dinner/dinner/cardentry.htm">Dinner-Diner Matching Probabilities</a>
%H <a href="/index/Ca#cardmatch">Index entries for sequences related to card matching</a>
%F 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 (5 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.
%e There are 32365 ways of matching exactly 2 cards when there are 2 cards of each kind and 5 kinds of card so T(2,2)=32365.
%p 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);
%p for n from 0 to 4 do seq(coeff(f(t,5,n),t,m)/n!^5,m=0..5*n); od;
%t 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, 5, n], t, m]/n!^5, {n, 0, 4}, {m, 0, 5*n}] // Flatten (* _Jean-François Alcover_, Oct 21 2013, after Maple *)
%Y Cf. A008290, A059056-A059071.
%K nonn,tabf,nice
%O 0,2
%A Barbara Haas Margolius (margolius(AT)math.csuohio.edu)