login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A059065 Card-matching numbers (Dinner-Diner 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 (list; graph; refs; listen; history; text; internal format)
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 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/((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: 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 18 03:46 EDT 2021. Contains 345098 sequences. (Running on oeis4.)