%I #16 Sep 21 2017 03:35:56

%S 1,9,8,6,0,1,4752,10752,11776,7680,3936,1024,384,0,16,17927568,

%T 64105344,109524960,117863424,89474544,49828608,21352896,6718464,

%U 1854576,279936,69984,0,1296,248341303296,1215287525376

%N Card-matching numbers (Dinner-Diner matching numbers).

%C This is a triangle of card matching numbers. Two decks each have 4 kinds of cards, n of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. 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)!.

%C rows are of length 1,5,9,13,...

%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 (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/((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 11776 ways of matching exactly 2 cards when there are 2 cards of each kind and 4 kinds of card so T(2,2)=11776.

%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 5 do seq(coeff(f(t,4,n),t,m),m=0..4*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, 4, n], t, m], {n, 0, 5}, {m, 0, 4*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)