A325004 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthotope using up to k colors.

1, 4, 1, 9, 6, 1, 16, 24, 10, 1, 25, 70, 57, 15, 1, 36, 165, 240, 126, 21, 1, 49, 336, 800, 730, 252, 28, 1, 64, 616, 2226, 3270, 2008, 462, 36, 1, 81, 1044, 5390, 11991, 11880, 5006, 792, 45, 1, 100, 1665, 11712, 37450, 56133, 38970, 11440, 1287, 55, 1

Offset: 1

Author

Robert A. Russell, Mar 23 2019

Keywords

• nonn
• tabl
• easy

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. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.

Also the number of oriented colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.

Examples

			Array begins with A(1,1):
1  4    9    16     25      36       49        64        81        100 ...
1  6   24    70    165     336      616      1044      1665       2530 ...
1 10   57   240    800    2226     5390     11712     23355      43450 ...
1 15  126   730   3270   11991    37450    102726    253485     573265 ...
1 21  252  2008  11880   56133   221725    756288   2283876    6228145 ...
1 28  462  5006  38970  235235  1161832   4873128  17838492   58208920 ...
1 36  792 11440 116400  894465  5495896  28162368 124122780  481650400 ...
1 45 1287 24310 319815 3114540 23739310 148116618 782798490 3596651740 ...
For A(1,2) = 4, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses two colors.
		

Links

• Robert A. Russell, Table of n, a(n) for n = 1..325
• Robin Chapman, answer to Coloring the faces of a hypercube, Math StackExchange, September 30, 2010.

Crossrefs

Cf. A325005 (unoriented), A325006 (chiral), A325007 (achiral), A325008 (exactly k colors)

Other n-dimensional polytopes: A324999 (simplex), A325012 (orthoplex)

Rows 1-3 are A000290, A006528, A047780.

Programs

Mathematica

Table[Binomial[Binomial[d-n+2,2]+n-1,n]+Binomial[Binomial[d-n+1,2],n],{d,1,11},{n,1,d}] // Flatten

Formula

A(n,k) = binomial(binomial(k+1,2) + n-1, n) + binomial(binomial(k,2),n).

A(n,k) = Sum_{j=1..2n} A325008(n,j) * binomial(k,j).

A(n,k) = A325005(n,k) + A325006(n,k) = 2*A325005(n,k) - A325007(n,k) = 2*A325006(n,k) + A325007(n,k).

G.f. for row n: Sum{j=1..2n} A325008(n,j) * x^j / (1-x)^(j+1).

Linear recurrence for row n: T(n,k) = Sum_{j=0..2n} binomial(-2-j,2n-j) * T(n,k-1-j).

G.f. for column k: 1/(1-x)^binomial(k+1,2) + (1+x)^binomial(k,2) - 2.