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 A059056 Penrice Christmas gift numbers, Card-matching numbers (Dinner-Diner matching numbers): Triangle T(n,k) = number of ways to get k matches for a deck with n cards, 2 of each kind. 22
 1, 0, 0, 1, 1, 0, 4, 0, 1, 10, 24, 27, 16, 12, 0, 1, 297, 672, 736, 480, 246, 64, 24, 0, 1, 13756, 30480, 32365, 21760, 10300, 3568, 970, 160, 40, 0, 1, 925705, 2016480, 2116836, 1418720, 677655, 243360, 67920, 14688, 2655, 320, 60, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS This is a triangle of card matching numbers. A deck has n kinds of cards, 2 of each kind. The deck is shuffled and dealt in to n hands with 2 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..2n). The probability of exactly k matches is T(n,k)/((2n)!/2^n). Rows are of length 1,3,5,7,... = A005408(n). [Edited by M. F. Hasler, Sep 21 2015] Analogous to A008290. - Zerinvary Lajos, Jun 10 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 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. 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 2) 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 4 ways of matching exactly 2 cards when there are 2 different kinds of cards, 2 of each in each of the two decks so T(2,2)=4. Triangle begins: 1 "0", 0, 1 1, '0', "4", 0, 1 10, 24, 27, '16', "12", 0, 1 297, 672, 736, 480, 246, '64', "24", 0, 1 13756, 30480, 32365, 21760, 10300, 3568, 970, '160', "40", 0, 1 925705, 2016480, 2116836, 1418720, 677655, 243360, 67920, 14688, 2655, '320', "60", 0, 1 Diagonal " ": T(n,2n-2) = 0, 4, 12, 24, 40, 60, 84, 112, 144, ... equals A046092 Diagonal ' ': T(n,2n-3) = 0, 16, 64, 160, 320, 560, 896, 1344, ... equals A102860 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 7 do seq(coeff(f(t, n, 2), t, m)/2^n, m=0..2*n); od; MATHEMATICA 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, 2]/2^n, t, m], {n, 0, 6}, {m, 0, 2*n}] // Flatten (* Jean-François Alcover, Sep 17 2012, translated from Maple *) CROSSREFS Cf. A059056-A059071, A008290. Sequence in context: A342372 A289222 A121301 * A344393 A127153 A178979 Adjacent sequences:  A059053 A059054 A059055 * A059057 A059058 A059059 KEYWORD nonn,tabf,nice AUTHOR Barbara Haas Margolius (margolius(AT)math.csuohio.edu) EXTENSIONS Additional comments from Zerinvary Lajos, Jun 18 2007 Edited by M. F. Hasler, Sep 21 2015 STATUS approved

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Last modified July 24 14:51 EDT 2021. Contains 346273 sequences. (Running on oeis4.)