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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.
9

%I #5 May 27 2019 17:54:22

%S 1,2,1,4,9,6,1,8,30,68,75,30,1,13,84,312,735,1020,735,210,1,19,192,

%T 1122,4155,10242,16380,15960,8505,1890,1,26,381,3322,18285,67679,

%U 173936,308056,363825,270900,114345,20790,1,34,687,8484,66765,352359,1305612,3479268,6668865,9035460,8378370,5031180,1756755,270270

%N 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.

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

%C Also the number of oriented colorings of the vertices of a regular n-dimensional orthoplex using exactly k colors.

%H Robert A. Russell, <a href="/A325008/b325008.txt">Table of n, a(n) for n = 1..132</a>

%F 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).

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

%e Triangle begins with T(1,1):

%e 1 2

%e 1 4 9 6

%e 1 8 30 68 75 30

%e 1 13 84 312 735 1020 735 210

%e 1 19 192 1122 4155 10242 16380 15960 8505 1890

%e 1 26 381 3322 18285 67679 173936 308056 363825 270900 114345 20790

%e 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.

%t 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

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

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

%K nonn,tabf,easy

%O 1,2

%A _Robert A. Russell_, May 27 2019