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A059068 Card-matching numbers (Dinner-Diner matching numbers).

%I A059068 #21 Aug 27 2024 14:30:54
%S A059068 1,9,8,6,0,1,297,672,736,480,246,64,24,0,1,13833,49464,84510,90944,
%T A059068 69039,38448,16476,5184,1431,216,54,0,1,748521,3662976,8607744,
%U A059068 12880512,13731616,11042688,6928704,3458432,1395126
%N A059068 Card-matching numbers (Dinner-Diner matching numbers).
%C A059068 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 j-th 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).
%C A059068 Rows have lengths 1,5,9,13,...
%C A059068 Analogous to A008290 - _Zerinvary Lajos_, Jun 22 2005
%D A059068 F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
%D A059068 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
%D A059068 R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
%H A059068 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 A059068 Barbara H. Margolius, <a href="http://academic.csuohio.edu/bmargolius/homepage/dinner/dinner/cardentry.htm">Dinner-Diner Matching Probabilities</a>
%H A059068 B. H. Margolius, <a href="http://www.jstor.org/stable/3219303">The Dinner-Diner Matching Problem</a>, Mathematics Magazine, 76 (2003), 107-118.
%H A059068 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 A059068 <a href="/index/Ca#cardmatch">Index entries for sequences related to card matching</a>
%F A059068 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 A059068 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.
%e A059068 Triangle begins:
%e A059068       1;
%e A059068       9,     8,     6,     0,     1;
%e A059068     297,   672,   736,   480,   246,    64,    24,    0,    1;
%e A059068   13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1;
%e A059068   ...
%p A059068 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 A059068 for n from 0 to 5 do seq(coeff(f(t,4,n),t,m)/n!^4,m=0..4*n); od;
%t A059068 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!^4, {n, 0, 4}, {m, 0, 4*n}] // Flatten (* _Jean-François Alcover_, Dec 17 2012, translated from Maple *)
%Y A059068 Cf. A008290, A059056-A059071.
%Y A059068 Cf. A008290.
%Y A059068 Row sums give A008977.
%K A059068 nonn,tabf,nice
%O A059068 0,2
%A A059068 Barbara Haas Margolius (margolius(AT)math.csuohio.edu)