A059065 Card-matching numbers (Dinner-Diner matching numbers).

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

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. 174-178.

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 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 (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/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the 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/((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 7 do seq(coeff(f(t, 2, n), t, m), m=0..2*n); od;

Mathematica

p[x_, k_] := k!^2*Sum[ x^j/((k-j)!^2*j!), {j, 0, k}];
f[t_, n_, k_] :=  Sum[ Coefficient[ p[x, k]^n, x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Table[ Coefficient[ f[t, 2, n], t, m], {n, 0, 7}, {m, 0, 2*n}] // Flatten (* Jean-François Alcover, Sep 17 2012, translated from Maple *)

Crossrefs

Cf. A008290, A059056-A059071.

Sequence in context: A095367 A368694 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