A325008 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.

1, 2, 1, 4, 9, 6, 1, 8, 30, 68, 75, 30, 1, 13, 84, 312, 735, 1020, 735, 210, 1, 19, 192, 1122, 4155, 10242, 16380, 15960, 8505, 1890, 1, 26, 381, 3322, 18285, 67679, 173936, 308056, 363825, 270900, 114345, 20790, 1, 34, 687, 8484, 66765, 352359, 1305612, 3479268, 6668865, 9035460, 8378370, 5031180, 1756755, 270270

Offset

1,2

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 exactly k colors.

Links

Robert A. Russell, Table of n, a(n) for n = 1..132

Formula

T(n,k) = Sum_{j=0..k-1} binomial(-j-2,k-j-1) * binomial(n + binomial(j+2,2)-1, n) + Sum_{j=0..k-2} binomial(j-k-1,j) * binomial(binomial(k-j,2),n).

T(n,k) = A325009(n,k) + A325010(n,k) = (A325009(n,k) + A325011(n,k)) / 2 = 2*A325010(n,k) + A325011(n,k).

Example

Triangle begins with T(1,1):
 1  2
 1  4   9    6
 1  8  30   68    75    30
 1 13  84  312   735  1020    735    210
 1 19 192 1122  4155 10242  16380  15960   8505   1890
 1 26 381 3322 18285 67679 173936 308056 363825 270900 114345 20790
For T(2,2)=4, there are two squares with just one edge for one color, one square with opposite edges the same color, and one square with opposite edges different colors.

Mathematica

Table[Sum[Binomial[-j-2, k-j-1] Binomial[n + Binomial[j+2, 2]-1, n], {j, 0, k-1}] + Sum[Binomial[j-k-1, j] Binomial[Binomial[k-j, 2], n], {j, 0, k-2}], {n, 1, 10}, {k, 1, 2n}] // Flatten

Crossrefs

Cf. A325009 (unoriented), A325010 (chiral), A325011 (achiral), A325004 (up to k colors).

Other n-dimensional polytopes: A325002 (simplex), A325016 (orthoplex).

Sequence in context: A077878 A128058 A163236 * A325016 A362984 A077160

Adjacent sequences: A325005 A325006 A325007 * A325009 A325010 A325011

Keyword

nonn,tabf,easy

Author

Robert A. Russell, May 27 2019

Status

approved