

A059063


Cardmatching numbers (DinnerDiner matching numbers).


0



1, 0, 0, 0, 0, 0, 120, 14400, 0, 360000, 0, 1440000, 0, 1440000, 0, 360000, 0, 14400, 3891456000, 26179200000, 83980800000, 171676800000, 249091200000, 270869184000, 226368000000, 150465600000, 77760000000
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OFFSET

0,7


COMMENTS

This is a triangle of card matching numbers. Two decks each have n kinds of cards, 5 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..5n). The probability of exactly k matches is T(n,k)/(5n)!.
rows are of length 1,6,11,16,...


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..26.
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 5) 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 of coefficient x^j of the rook polynomial.


EXAMPLE

There are 360,000 ways of matching exactly 2 cards when there are 2 different kinds of cards, 5 of each in each of the two decks so T(2,2)=360,000.


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, 5), t, m), m=0..5*n); od;


MATHEMATICA

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[Coefficient[r[x, n, k], x, j]*(t1)^j*(n*kj)!, {j, 0, n*k}]; k = 5; Table[ Table[ Coefficient[f[t, n, k], t, m], {m, 0, k*n}], {n, 0, 4}] // Flatten (* JeanFrançois Alcover, Oct 21 2013, after Maple *)


CROSSREFS

Cf. A008290, A059056A059071.
Sequence in context: A104592 A135379 A296913 * A224178 A223870 A224390
Adjacent sequences: A059060 A059061 A059062 * A059064 A059065 A059066


KEYWORD

nonn,tabf,nice


AUTHOR

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


STATUS

approved



