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%I #26 Nov 08 2020 12:32:49
%S 1,0,0,0,0,1,1,0,16,0,36,0,16,0,1,346,1824,4536,7136,7947,6336,3936,
%T 1728,684,128,48,0,1,748521,3662976,8607744,12880512,13731616,
%U 11042688,6928704,3458432,1395126,453888,122016,25344,4824,512,96,0,1,3993445276
%N Card-matching numbers (Dinner-Diner matching numbers).
%C This is a triangle of card matching numbers. A deck has n kinds of cards, 4 of each kind. The deck is shuffled and dealt in to n hands with 4 cards each. 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)!/(4!)^n).
%C Rows have lengths 1,5,9,13,...
%C Analogous to A008290 - _Zerinvary Lajos_, Jun 22 2005
%D F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
%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 Barbara H. Margolius, <a href="http://academic.csuohio.edu/bmargolius/homepage/dinner/dinner/cardentry.htm">Dinner-Diner Matching Probabilities</a>
%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 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 <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, k is the number of cards of each kind (here k is 4) 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 There are 16 ways of matching exactly 2 cards when there are 2 different kinds of cards, 4 of each so T(2,2)=16.
%e From _Joerg Arndt_, Nov 08 2020: (Start)
%e The first few rows are
%e 1
%e 0, 0, 0, 0, 1
%e 1, 0, 16, 0, 36, 0, 16, 0, 1
%e 346, 1824, 4536, 7136, 7947, 6336, 3936, 1728, 684, 128, 48, 0, 1
%e 748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126, 453888, 122016, 25344, 4824, 512, 96, 0, 1
%e 3993445276, 18743463360, 42506546320, 61907282240, 64917874125, 52087325696, 33176621920, 17181584640, 7352761180, 2628808000, 790912656, 201062080, 43284010, 7873920, 1216000, 154496, 17640, 1280, 160, 0, 1 (End)
%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,n,4),t,m)/4!^n,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, n, 4], t, m]/4!^n, {n, 0, 5}, {m, 0, 4*n}] // Flatten (* _Jean-François Alcover_, Feb 22 2013, translated from Maple *)
%Y Cf. A008290, A059056-A059071.
%K nonn,tabf,nice
%O 0,9
%A Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
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