

A059059


Cardmatching numbers (DinnerDiner matching numbers).


0



1, 0, 0, 0, 6, 36, 0, 324, 0, 324, 0, 36, 12096, 46656, 81648, 93960, 69984, 40824, 11664, 5832, 0, 216, 17927568, 64105344, 109524960, 117863424, 89474544, 49828608, 21352896, 6718464, 1854576, 279936, 69984, 0, 1296
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OFFSET

0,5


COMMENTS

This is a triangle of card matching numbers. Two decks each have n kinds of cards, 3 of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. 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)!.
Rows are of length 1,4,7,10,...


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. 174178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.


LINKS

Table of n, a(n) for n=0..34.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a CardMatching Problem, Mathematics Magazine 69 (1996), 190197.
Barbara H. Margolius, DinnerDiner Matching Probabilities
B. H. Margolius, The DinnerDiner Matching Problem, Mathematics Magazine, 76 (2003), 107118.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617620.
Index entries for sequences related to card matching


FORMULA

G.f.: sum(coeff(R(x, n, k), x, j)*(t1)^j*(n*kj)!, 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/((kj)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the jth coefficient on x of the rook polynomial.


EXAMPLE

There are 324 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)=324.


MAPLE

p := (x, k)>k!^2*sum(x^j/((kj)!^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)*(t1)^j*(n*kj)!, j=0..n*k);
for n from 0 to 4 do seq(coeff(f(t, n, 3), t, m), m=0..3*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, 3], t, m], {n, 0, 4}, {m, 0, 3*n}] // Flatten (* JeanFrançois Alcover, Oct 21 2013, after Maple *)


CROSSREFS

Cf. A008290, A059056A059071.
Sequence in context: A193000 A193001 A128298 * A304255 A050112 A250202
Adjacent sequences: A059056 A059057 A059058 * A059060 A059061 A059062


KEYWORD

nonn,tabf,nice


AUTHOR

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


STATUS

approved



