

A059065


Cardmatching numbers (DinnerDiner matching numbers).


0



1, 1, 0, 1, 4, 0, 16, 0, 4, 36, 0, 324, 0, 324, 0, 36, 576, 0, 9216, 0, 20736, 0, 9216, 0, 576, 14400, 0, 360000, 0, 1440000, 0, 1440000, 0, 360000, 0, 14400, 518400, 0, 18662400, 0, 116640000, 0, 207360000, 0, 116640000
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OFFSET

0,5


COMMENTS

This is a triangle of card matching numbers. Two decks each have 2 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..2n). The probability of exactly k matches is T(n,k)/(2n)!.
rows are of length 1,3,5,7,...


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..44.
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 (2 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 of coefficient x^j of the rook polynomial.


EXAMPLE

There are 16 ways of matching exactly 2 cards when there are 2 cards of each kind and 2 kinds of card so T(2,2)=16.


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 7 do seq(coeff(f(t, 2, n), t, m), m=0..2*n); od;


MATHEMATICA

p[x_, k_] := k!^2*Sum[ x^j/((kj)!^2*j!), {j, 0, k}];
f[t_, n_, k_] := Sum[ Coefficient[ p[x, k]^n, x, j]*(t1)^j*(n*kj)!, {j, 0, n*k}]; Table[ Coefficient[ f[t, 2, n], t, m], {n, 0, 7}, {m, 0, 2*n}] // Flatten (* JeanFrançois Alcover, Sep 17 2012, translated from Maple *)


CROSSREFS

Cf. A008290, A059056A059071.
Sequence in context: A081162 A095367 A060052 * A170771 A170772 A079986
Adjacent sequences: A059062 A059063 A059064 * A059066 A059067 A059068


KEYWORD

nonn,tabf,nice


AUTHOR

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


STATUS

approved



