A325005 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthotope using up to k colors.

1, 3, 1, 6, 6, 1, 10, 21, 10, 1, 15, 55, 56, 15, 1, 21, 120, 220, 126, 21, 1, 28, 231, 680, 715, 252, 28, 1, 36, 406, 1771, 3060, 2002, 462, 36, 1, 45, 666, 4060, 10626, 11628, 5005, 792, 45, 1, 55, 1035, 8436, 31465, 53130, 38760, 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 unoriented colorings are the same if congruent; chiral pairs are counted as one.

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

Examples

			Array begins with A(1,1):
1  3    6    10     15      21       28        36        45         55 ...
1  6   21    55    120     231      406       666      1035       1540 ...
1 10   56   220    680    1771     4060      8436     16215      29260 ...
1 15  126   715   3060   10626    31465     82251    194580     424270 ...
1 21  252  2002  11628   53130   201376    658008   1906884    5006386 ...
1 28  462  5005  38760  230230  1107568   4496388  15890700   50063860 ...
1 36  792 11440 116280  888030  5379616  26978328 115775100  436270780 ...
1 45 1287 24310 319770 3108105 23535820 145008513 752538150 3381098545 ...
For A(1,2) = 3, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses one color for each vertex.
		

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. A325004 (oriented), A325006 (chiral), A325007 (achiral), A325009 (exactly k colors).

Other n-dimensional polytopes: A325000 (simplex), A325013 (orthoplex).

Rows 1-3 are A000217, A002817, A198833.

Programs

Mathematica

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

Formula

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

A(n,k) = Sum_{j=1..2n} A325009(n,j) * binomial(k,j).

A(n,k) = A325004(n,k) - A325006(n,k) = (A325004(n,k) + A325007(n,k)) / 2 = A325006(n,k) + A325007(n,k).

G.f. for row n: Sum_{j=1..2n} A325009(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.