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A059056
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Penrice Christmas gift numbers, Card-matching numbers (Dinner-Diner matching numbers).
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22
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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;
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OFFSET
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0,7
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COMMENTS
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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,...
Analogous to A008290. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 10 2005
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REFERENCES
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F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
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.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 174-178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
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LINKS
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Table of n, a(n) for n=0..48.
Index entries for sequences related to card matching
Barbara H. Margolius, Dinner-Diner Matching Probabilities
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FORMULA
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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.
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EXAMPLE
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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
No. 1. diagonal : " " :0, 4, 12, 24, 40, 60, 84, 112, 144, 180, equals A046092
No. 2. diagonal : ' ' :0, 16, 64, 160, 320, 560, 896, 1344, equals A102860
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MAPLE
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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;
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A059056-A059071, A008290.
Sequence in context: A199786 A189245 A121301 * A127153 A178979 A189355
Adjacent sequences: A059053 A059054 A059055 * A059057 A059058 A059059
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
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EXTENSIONS
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Additional comments from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 18 2007
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STATUS
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approved
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