1,3

Card-matching numbers (Dinner-Diner matching numbers): A deck has n kinds of cards, 2 of each kind. The deck is shuffled and dealt in to n hands with 2 cards each. A match occurs for every card in the j-th hand of kind j. A(n) is the number of ways of achieving no matches. The probability of no matches is A(n)/((2n)!/2!^n).

Number of permutations of (0 to infinity) distinct letters (ABC......XYW etc.) each with 2 copies such that free (0) remain fixed points. E.g. if AABBCCDDEEFF (2*6=12 letters) then free (0) fixed points n5=925705 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 02 2006

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.

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

B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 187.

T. D. Noe, Table of n, a(n) for n=1..100

Index entries for sequences related to card matching

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, k is the number of cards of each kind (2 in this case) 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 Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.

a[n]=n(2n-1)*a[n-1]+2n(n-1)a[n-2]-(2n-1), a[1]=0, a[2]=1

There are 297 ways of achieving zero matches when there are 2 cards of each kind and 4 kinds of card so A(4)=297.

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); seq(f(0, n, 2)/2!^n, n=0..18);

Equals A000316/2^n.

Cf. A008290, A059056-A059071, A033581.

See A059072 for another version.

Sequence in context: A049387 A024295 A059072 this_sequence A125288 A069672 A129005

Adjacent sequences: A000456 A000457 A000458 this_sequence A000460 A000461 A000462

nonn,nice,easy

njas

Formulae, more terms, etc. from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 22 2000