A059067 Card-matching numbers (Dinner-Diner matching numbers).

1, 2, 3, 0, 1, 80, 192, 216, 128, 96, 0, 8, 12096, 46656, 81648, 93960, 69984, 40824, 11664, 5832, 0, 216, 4783104, 25214976, 62705664, 98648064, 109859328, 87588864, 54411264, 23887872, 9455616, 1769472, 663552, 0

Offset

0,2

Comments

This is a triangle of card matching numbers. Two decks each have 3 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..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. 174-178.

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

Links

Table of n, a(n) for n=0..33.

F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.

Barbara H. Margolius, Dinner-Diner Matching Probabilities

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.

Index entries for sequences related to card matching

Formula

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 (3 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/((k-j)!^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 216 ways of matching exactly 2 cards when there are 2 cards of each kind and 3 kinds of card so T(2,2)=216.

Maple

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 5 do seq(coeff(f(t, 3, n), 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, 3, n], t, m], {n, 0, 5}, {m, 0, 3*n}] // Flatten (* Jean-François Alcover, Oct 21 2013, after Maple *)

Crossrefs

Cf. A008290, A059056-A059071.

Sequence in context: A008290 A322147 A059066 * A356707 A352938 A065861

Adjacent sequences: A059064 A059065 A059066 * A059068 A059069 A059070

Keyword

nonn,tabf,nice

Author

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

Status

approved