A059058 - OEIS

%I #23 Sep 26 2017 05:08:24

%S 1,0,0,0,1,1,0,9,0,9,0,1,56,216,378,435,324,189,54,27,0,1,13833,49464,

%T 84510,90944,69039,38448,16476,5184,1431,216,54,0,1,6699824,23123880,

%U 38358540,40563765,30573900,17399178,7723640

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

%C This is a triangle of card matching numbers. A deck has n kinds of cards, 3 of each kind. The deck is shuffled and dealt in to n hands with 3 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..3n). The probability of exactly k matches is T(n,k)/((3n)!/(3!)^n).

%C Rows have lengths 1,4,7,10,...

%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 Vincenzo Librandi, <a href="/A059058/b059058.txt">Rows n = 1..30, flattened</a>

%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 3) 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 9 ways of matching exactly 2 cards when there are 2 different kinds of cards, 3 of each in each of the two decks so T(2,2)=9.

%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 6 do seq(coeff(f(t,n,3),t,m)/3!^n,m=0..3*n); od;

%t p[x_, k_] := k!^2*Sum[ x^j/((k-j)!^2*j!), {j, 0, k}]; f[t_, n_, k_] := Sum[ Coefficient[ p[x, k]^n, x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Flatten[ Table[ Coefficient[ f[t, n, 3], t, m]/3!^n, {n, 0, 6}, {m, 0, 3n}]] (* _Jean-François Alcover_, Jan 31 2012, after Maple *)

%Y Cf. A008290, A059056-A059071, A008290.

%K nonn,tabf,nice

%O 0,8

%A Barbara Haas Margolius (margolius(AT)math.csuohio.edu)