|
| |
|
|
A079908
|
|
Solution to the Dancing School Problem with 3 girls and n+3 boys: f(3,n).
|
|
36
|
|
|
|
1, 4, 14, 36, 76, 140, 234, 364, 536, 756, 1030, 1364, 1764, 2236, 2786, 3420, 4144, 4964, 5886, 6916, 8060, 9324, 10714, 12236, 13896, 15700, 17654, 19764, 22036, 24476, 27090, 29884, 32864, 36036, 39406, 42980, 46764, 50764, 54986, 59436
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
The Dancing School Problem: a line of g girls (g>0) with integer heights ranging from m to m+g-1 cm and a line of g+h boys (h>=0) ranging in height from m to m+g+h-1 cm are facing each other in a dancing school (m is the minimal height of both girls and boys).
A girl of height l can choose a boy of her own height or taller with a maximum of l+h cm. We call the number of possible matchings f(g,h).
This problem is equivalent to a rooks problem: The number of possible placings of g non-attacking rooks on a g X g+h chessboard with the restriction i <= j <= i+h for the placement of a rook on square (i,j): f(g,h) = per(B), the permanent of the corresponding (0,1)-matrix B, b(i, j)=1 if and only if i <= j <= i+h
f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h.
For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = max(1, n^3 + 3*n), found by elementary counting.
G.f.: 1+2*x*(2-x+2*x^2)/(1-x)^4. - R. J. Mathar, Nov 19 2007
|
|
|
MATHEMATICA
|
|
|
|
PROG
|
(PARI) concat(1, vector(30, n, n^3+3*n)) \\ Derek Orr, Jun 18 2015
|
|
|
CROSSREFS
|
Cf. Essentially the same as A061989.
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
