|
%I #18 Oct 20 2020 08:17:06
%S 1,4,1,9,4,1,16,11,5,1,25,24,15,6,1,36,45,36,21,7,1,49,76,75,56,28,8,
%T 1,64,119,141,127,84,36,9,1,81,176,245,258,210,120,45,10,1,100,249,
%U 400,483,463,330,165,55,11,1,121,340,621,848,931,792,495,220,66,12,1
%N Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.
%C For n=1, the figure is a line segment with two vertices. For n=2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with four triangular faces. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Each of its n+1 facets is a regular (n-1)-dimensional simplex. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
%H Robert A. Russell, <a href="/A324999/b324999.txt">Table of n, a(n) for n = 1..1275</a>
%F A(n,k) = binomial(n+k,n+1) + binomial(k,n+1).
%F A(n,k) = Sum_{j=1..n+1} A325002(n,j) * binomial(k,j).
%F A(n,k) = A325000(n,k) + A325000(n,k-n) = 2*A325000(n,k) - A325001(n,k) = 2*A325000(n,k-n) + A325001(n,k).
%F G.f. for row n: (x + x^(n+1)) / (1-x)^(n+2).
%F Linear recurrence for row n: A(n,k) = Sum_{j=1..n+2} -binomial(j-n-3,j) * A(n,k-j).
%F G.f. for column k: (1 - 2*(1-x)^k + (1-x^2)^k) / (x*(1-x)^k) - 2*k.
%e The array begins with A(1,1):
%e 1 4 9 16 25 36 49 64 81 100 121 144 169 196 ...
%e 1 4 11 24 45 76 119 176 249 340 451 584 741 924 ...
%e 1 5 15 36 75 141 245 400 621 925 1331 1860 2535 3381 ...
%e 1 6 21 56 127 258 483 848 1413 2254 3465 5160 7475 10570 ...
%e 1 7 28 84 210 463 931 1744 3087 5215 8470 13300 20280 30135 ...
%e 1 8 36 120 330 792 1717 3440 6471 11560 19778 32616 52104 80952 ...
%e 1 9 45 165 495 1287 3003 6436 12879 24355 43923 76077 127257 206493 ...
%e 1 10 55 220 715 2002 5005 11440 24311 48630 92433 168180 294645 499422 ...
%e ...
%e For A(1,2) = 4, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses two colors. For A(2,2)=4, the triangle may have 0, 1, 2, or 3 edges of one color.
%t Table[Binomial[d+1,n+1] + Binomial[d+1-n,n+1], {d,1,15}, {n,1,d}] // Flatten
%Y Cf. A325000 (unoriented), A325000(n,k-n) (chiral), A325001 (achiral), A325002 (exactly k colors), A327083 (edges, ridges), A337883 (faces, peaks), A325004 (orthotope facets, orthoplex vertices), A325012 (orthoplex facets, orthotope vertices).
%Y Rows 1-4 are A000290, A006527, A006008, A337895.
%K nonn,tabl
%O 1,2
%A _Robert A. Russell_, Mar 23 2019
|
