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 A059058 Card-matching numbers (Dinner-Diner matching numbers). 7
 1, 0, 0, 0, 1, 1, 0, 9, 0, 9, 0, 1, 56, 216, 378, 435, 324, 189, 54, 27, 0, 1, 13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1, 6699824, 23123880, 38358540, 40563765, 30573900, 17399178, 7723640 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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). Rows have lengths 1,4,7,10,... Analogous to A008290 - Zerinvary Lajos, Jun 22 2005 REFERENCES F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12. J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178. R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71. LINKS Vincenzo Librandi, Rows n = 1..30, flattened F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197. Barbara H. Margolius, Dinner-Diner Matching Probabilities B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118. S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620. Index entries for sequences related to card matching FORMULA 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. EXAMPLE 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. MAPLE 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); for n from 0 to 6 do seq(coeff(f(t, n, 3), t, m)/3!^n, m=0..3*n); od; MATHEMATICA 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 *) CROSSREFS Cf. A008290, A059056-A059071, A008290. Sequence in context: A086199 A167545 A272965 * A343587 A021015 A372947 Adjacent sequences: A059055 A059056 A059057 * A059059 A059060 A059061 KEYWORD nonn,tabf,nice AUTHOR Barbara Haas Margolius (margolius(AT)math.csuohio.edu) STATUS approved

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Last modified August 6 07:30 EDT 2024. Contains 374960 sequences. (Running on oeis4.)