A324999 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.

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, 1, 64, 119, 141, 127, 84, 36, 9, 1, 81, 176, 245, 258, 210, 120, 45, 10, 1, 100, 249, 400, 483, 463, 330, 165, 55, 11, 1, 121, 340, 621, 848, 931, 792, 495, 220, 66, 12, 1

Offset

1,2

Comments

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.

Links

Robert A. Russell, Table of n, a(n) for n = 1..1275

Formula

A(n,k) = binomial(n+k,n+1) + binomial(k,n+1).

A(n,k) = Sum_{j=1..n+1} A325002(n,j) * binomial(k,j).

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).

G.f. for row n: (x + x^(n+1)) / (1-x)^(n+2).

Linear recurrence for row n: A(n,k) = Sum_{j=1..n+2} -binomial(j-n-3,j) * A(n,k-j).

G.f. for column k: (1 - 2*(1-x)^k + (1-x^2)^k) / (x*(1-x)^k) - 2*k.

Example

The array begins with A(1,1):
  1  4  9  16  25   36   49    64    81   100   121    144    169    196 ...
  1  4 11  24  45   76  119   176   249   340   451    584    741    924 ...
  1  5 15  36  75  141  245   400   621   925  1331   1860   2535   3381 ...
  1  6 21  56 127  258  483   848  1413  2254  3465   5160   7475  10570 ...
  1  7 28  84 210  463  931  1744  3087  5215  8470  13300  20280  30135 ...
  1  8 36 120 330  792 1717  3440  6471 11560 19778  32616  52104  80952 ...
  1  9 45 165 495 1287 3003  6436 12879 24355 43923  76077 127257 206493 ...
  1 10 55 220 715 2002 5005 11440 24311 48630 92433 168180 294645 499422 ...
  ...
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.

Mathematica

Table[Binomial[d+1, n+1] + Binomial[d+1-n, n+1], {d, 1, 15}, {n, 1, d}] // Flatten

Crossrefs

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).

Rows 1-4 are A000290, A006527, A006008, A337895.

Sequence in context: A339635 A085691 A055461 * A104796 A132020 A175643

Adjacent sequences: A324996 A324997 A324998 * A325000 A325001 A325002

Keyword

nonn,tabl

Author

Robert A. Russell, Mar 23 2019

Status

approved