A325007 - OEIS

%I #12 Oct 01 2019 07:16:06

%S 1,2,1,3,6,1,4,18,10,1,5,40,55,15,1,6,75,200,126,21,1,7,126,560,700,

%T 252,28,1,8,196,1316,2850,1996,462,36,1,9,288,2730,9261,11376,5004,

%U 792,45,1,10,405,5160,25480,50127,38550,11440,1287,55,1,11,550,9075,61776,181027,225225,116160,24310,2002,66,1

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

%C 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. An achiral coloring is identical to its reflection.

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

%H Robert A. Russell, <a href="/A325007/b325007.txt">Table of n, a(n) for n = 1..325</a>

%H Robin Chapman, answer to <a href="https://math.stackexchange.com/q/5732/">Coloring the faces of a hypercube</a>, Math StackExchange, September 30, 2010.

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

%F A(n,k) = Sum_{j=1..2*n} A325011(n,j) * binomial(k,j).

%F A(n,k) = 2*A325005(n,k) - A325004(n,k) = (A325004(n,k) - 2*A325006(n,k)) / 2 = A325005(n,k) + A325006(n,k).

%F G.f. for row n: Sum{j=1..2*n} A325011(n,j) * x^j / (1-x)^(j+1).

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

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

%e Array begins with A(1,1):

%e 1  2   3     4      5      6       7        8         9        10 ...

%e 1  6  18    40     75    126     196      288       405       550 ...

%e 1 10  55   200    560   1316    2730     5160      9075     15070 ...

%e 1 15 126   700   2850   9261   25480    61776    135675    275275 ...

%e 1 21 252  1996  11376  50127  181027   559728   1529892   3784627 ...

%e 1 28 462  5004  38550 225225 1053304  4119648  13942908  41918800 ...

%e 1 36 792 11440 116160 881595 5263336 25794288 107427420 390891160 ...

%e For a(2,2)=6, all colorings are achiral: two with just one of the colors, two with one color on just one edge, one with opposite colors the same, and one with opposite colors different.

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

%o (PARI) a(n, k) = binomial(binomial(k+1, 2)+n-1, n) - binomial(binomial(k, 2), n)

%o array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))

%o /* Print initial 6 rows and 8 columns of array as follows: */

%o array(6, 8) \\ _Felix Fröhlich_, May 30 2019

%Y Cf. A325004 (oriented), A325005 (unoriented), A325006 (chiral), A325011 (exactly k colors).

%Y Other n-dimensional polytopes: A325001 (simplex), A325015 (orthoplex).

%Y Rows 1-2 are A000027, A002411; column 2 is A186783(n+2).

%K nonn,tabl,easy

%O 1,2

%A _Robert A. Russell_, May 27 2019