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%I #21 Apr 12 2023 10:52:47
%S 1,1,0,1,1,0,4,0,1,1,0,9,0,9,0,1,1,0,16,0,36,0,16,0,1,1,0,25,0,100,0,
%T 100,0,25,0,1,1,0,36,0,225,0,400,0,225,0,36,0,1,1,0,49,0,441,0,1225,0,
%U 1225,0,441,0,49,0,1,1,0,64,0,784,0,3136,0,4900,0
%N Card-matching numbers (Dinner-Diner matching numbers).
%C This is a triangle of card matching numbers. A deck has 2 kinds of cards, n of each kind. The deck is shuffled and dealt in to 2 hands 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..2n). An odd number of matches is impossible, so alternating elements in each row of the triangle are zero. The probability of exactly k matches is T(n,k)/((2n)!/n!^2).
%C Rows have lengths 1,3,5,7,...
%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="https://web.archive.org/web/20201128023513/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 (2 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 x^j of the rook polynomial.
%e There are 4 ways of matching exactly 2 cards when there are 2 cards of each kind and 2 kinds of card so T(2,2)=4.
%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 10 do seq(coeff(f(t,2,n),t,m)/n!^2,m=0..2*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[ Table[ Coefficient[f[t, 2, n], t, m]/n!^2, {m, 0, 2n}], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jan 25 2013, translated from Maple *)
%Y Cf. A008290, A059056-A059071.
%Y Cf. A008290.
%K nonn,tabf,nice
%O 0,7
%A Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
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