|
| |
|
|
A325006
|
|
Array read by descending antidiagonals: A(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthotope using up to k colors.
|
|
12
|
|
|
|
0, 1, 0, 3, 0, 0, 6, 3, 0, 0, 10, 15, 1, 0, 0, 15, 45, 20, 0, 0, 0, 21, 105, 120, 15, 0, 0, 0, 28, 210, 455, 210, 6, 0, 0, 0, 36, 378, 1330, 1365, 252, 1, 0, 0, 0, 45, 630, 3276, 5985, 3003, 210, 0, 0, 0, 0, 55, 990, 7140, 20475, 20349, 5005, 120, 0, 0, 0, 0, 66, 1485, 14190, 58905, 98280, 54264, 6435, 45, 0, 0, 0, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,4
|
|
|
COMMENTS
|
Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. The chiral colorings of its facets come in pairs, each the reflection of the other.
Also the number of chiral pairs of colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.
|
|
|
LINKS
|
|
|
|
FORMULA
|
A(n,k) = binomial(binomial(k,2),n).
A(n,k) = Sum_{j=1..2*n} A325010(n,j) * binomial(k,j).
G.f. for row n: Sum{j=1..2*n} A325010(n,j) * x^j / (1-x)^(j+1).
Linear recurrence for row n: T(n,k) = Sum_{j=0..2*n} binomial(-2-j,2*n-j) * T(n,k-1-j).
G.f. for column k: (1+x)^binomial(k,2) - 1.
|
|
|
EXAMPLE
|
Array begins with A(1,1):
0 1 3 6 10 15 21 28 36 45 55 ...
0 0 3 15 45 105 210 378 630 990 1485 ...
0 0 1 20 120 455 1330 3276 7140 14190 26235 ...
0 0 0 15 210 1365 5985 20475 58905 148995 341055 ...
0 0 0 6 252 3003 20349 98280 376992 1221759 3478761 ...
0 0 0 1 210 5005 54264 376740 1947792 8145060 28989675 ...
0 0 0 0 120 6435 116280 1184040 8347680 45379620 202927725 ...
0 0 0 0 45 6435 203490 3108105 30260340 215553195 1217566350 ...
0 0 0 0 10 5005 293930 6906900 94143280 886163135 6358402050 ...
0 0 0 0 1 3003 352716 13123110 254186856 3190187286 29248649430 ...
For a(2,3)=3, each chiral pair consists of two adjacent edges of the square with one of the three colors.
|
|
|
MATHEMATICA
|
Table[Binomial[Binomial[d-n+1, 2], n], {d, 1, 12}, {n, 1, d}] // Flatten
|
|
|
PROG
|
(PARI) a(n, k) = binomial(binomial(k, 2), n)
array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))
/* Print initial 10 rows and 11 columns of array as follows: */
|
|
|
CROSSREFS
|
Other n-dimensional polytopes: A007318(k,n+1) (simplex), A325014 (orthoplex)
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
