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%I #12 Oct 01 2019 07:16:37
%S 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,
%T 28,210,455,210,6,0,0,0,36,378,1330,1365,252,1,0,0,0,45,630,3276,5985,
%U 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
%N 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.
%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. The chiral colorings of its facets come in pairs, each the reflection of the other.
%C Also the number of chiral pairs of colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.
%H Robert A. Russell, <a href="/A325006/b325006.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,2),n).
%F A(n,k) = Sum_{j=1..2*n} A325010(n,j) * binomial(k,j).
%F A(n,k) = A325004(n,k) - A325005(n,k) = (A325004(n,k) - A325007(n,k)) / 2 = A325005(n,k) - A325007(n,k).
%F G.f. for row n: Sum{j=1..2*n} A325010(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+x)^binomial(k,2) - 1.
%e Array begins with A(1,1):
%e 0 1 3 6 10 15 21 28 36 45 55 ...
%e 0 0 3 15 45 105 210 378 630 990 1485 ...
%e 0 0 1 20 120 455 1330 3276 7140 14190 26235 ...
%e 0 0 0 15 210 1365 5985 20475 58905 148995 341055 ...
%e 0 0 0 6 252 3003 20349 98280 376992 1221759 3478761 ...
%e 0 0 0 1 210 5005 54264 376740 1947792 8145060 28989675 ...
%e 0 0 0 0 120 6435 116280 1184040 8347680 45379620 202927725 ...
%e 0 0 0 0 45 6435 203490 3108105 30260340 215553195 1217566350 ...
%e 0 0 0 0 10 5005 293930 6906900 94143280 886163135 6358402050 ...
%e 0 0 0 0 1 3003 352716 13123110 254186856 3190187286 29248649430 ...
%e For a(2,3)=3, each chiral pair consists of two adjacent edges of the square with one of the three colors.
%t Table[Binomial[Binomial[d-n+1,2],n],{d,1,12},{n,1,d}] // Flatten
%o (PARI) a(n, k) = 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 10 rows and 11 columns of array as follows: */
%o array(10, 11) \\ _Felix Fröhlich_, May 30 2019
%Y Cf. A325004 (oriented), A325005 (unoriented), A325007 (achiral), A325010 (exactly k colors)
%Y Other n-dimensional polytopes: A007318(k,n+1) (simplex), A325014 (orthoplex)
%Y Rows 1-3 are A161680, A050534, A093566(n+1), A234249(n-1)
%K nonn,tabl,easy
%O 1,4
%A _Robert A. Russell_, May 27 2019
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