

A059069


Cardmatching numbers (DinnerDiner matching numbers).


0



1, 9, 8, 6, 0, 1, 4752, 10752, 11776, 7680, 3936, 1024, 384, 0, 16, 17927568, 64105344, 109524960, 117863424, 89474544, 49828608, 21352896, 6718464, 1854576, 279936, 69984, 0, 1296, 248341303296, 1215287525376
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

This is a triangle of card matching numbers. Two decks each have 4 kinds of cards, n 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..4n). The probability of exactly k matches is T(n,k)/(4n)!.
rows are of length 1,5,9,13,...


REFERENCES

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617620.
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..29.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a CardMatching Problem, Mathematics Magazine 69 (1996), 190197.
B. H. Margolius, The DinnerDiner Matching Problem, Mathematics Magazine, 76 (2003), 107118.
Barbara H. Margolius, DinnerDiner Matching Probabilities
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 (4 in this case), k is the number of cards of each kind 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 coefficient for x^j of the rook polynomial.


EXAMPLE

There are 11776 ways of matching exactly 2 cards when there are 2 cards of each kind and 4 kinds of card so T(2,2)=11776.


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 5 do seq(coeff(f(t, 4, n), t, m), m=0..4*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, 4, n], t, m], {n, 0, 5}, {m, 0, 4*n}] // Flatten (* JeanFrançois Alcover, Oct 21 2013, after Maple *)


CROSSREFS

Cf. A008290, A059056A059071.
Sequence in context: A155791 A327341 A059068 * A084660 A002391 A193626
Adjacent sequences: A059066 A059067 A059068 * A059070 A059071 A059072


KEYWORD

nonn,tabf,nice


AUTHOR

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


STATUS

approved



