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A059057 Penrice Christmas gift numbers, Card-matching numbers (Dinner-Diner matching numbers).

%I A059057 #17 Sep 26 2017 05:08:17
%S A059057 1,0,0,2,4,0,16,0,4,80,192,216,128,96,0,8,4752,10752,11776,7680,3936,
%T A059057 1024,384,0,16,440192,975360,1035680,696320,329600,114176,31040,5120,
%U A059057 1280,0,32,59245120,129054720,135477504,90798080
%N A059057 Penrice Christmas gift numbers, Card-matching numbers (Dinner-Diner matching numbers).
%C A059057 This is a triangle of card matching numbers. Two decks each have n kinds of cards, 2 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..2n). The probability of exactly k matches is T(n,k)/(2n)!.
%C A059057 Rows are of length 1,3,5,7,...
%D A059057 F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
%D A059057 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
%D A059057 R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
%H A059057 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 A059057 Barbara H. Margolius, <a href="http://academic.csuohio.edu/bmargolius/homepage/dinner/dinner/cardentry.htm">Dinner-Diner Matching Probabilities</a>
%H A059057 B. H. Margolius, <a href="http://www.jstor.org/stable/3219303">The Dinner-Diner Matching Problem</a>, Mathematics Magazine, 76 (2003), 107-118.
%H A059057 S. G. Penrice, <a href="http://www.jstor.org/stable/2324927">Derangements, permanents and Christmas presents</a>, The American Mathematical Monthly 98(1991), 617-620.
%H A059057 <a href="/index/Ca#cardmatch">Index entries for sequences related to card matching</a>
%F A059057 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, k is the number of cards of each kind (here k is 2) 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 j-th coefficient on x of the rook polynomial.
%e A059057 There are 16 ways of matching exactly 2 cards when there are 2 different kinds of cards, 2 of each in each of the two decks so T(2,2)=16.
%p A059057 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 A059057 for n from 0 to 6 do seq(coeff(f(t,n,2),t,m),m=0..2*n); od;
%t A059057 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}]; Flatten[ Table[ Coefficient[ f[t, n, 2], t, m], {n, 0, 6}, {m, 0, 2 n}]](* _Jean-François Alcover_, Nov 28 2011, translated from Maple *)
%Y A059057 Cf. A008290, A059056-A059071.
%K A059057 nonn,tabf,nice
%O A059057 0,4
%A A059057 Barbara Haas Margolius (margolius(AT)math.csuohio.edu)