A325000 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.

1, 3, 1, 6, 4, 1, 10, 10, 5, 1, 15, 20, 15, 6, 1, 21, 35, 35, 21, 7, 1, 28, 56, 70, 56, 28, 8, 1, 36, 84, 126, 126, 84, 36, 9, 1, 45, 120, 210, 252, 210, 120, 45, 10, 1, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 66, 220, 495, 792, 924, 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 unoriented colorings are the same if congruent; chiral pairs are counted as one.

Note that antidiagonals are part of rows of the Pascal triangle.

T(n,k-n) is the number of chiral pairs of colorings of the facets (or vertices) of a regular n-dimensional simplex using k or fewer colors. - Robert A. Russell, Sep 28 2020

Links

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

Formula

T(n,k) = binomial(n+k,n+1) = A007318(n+k,n+1).

T(n,k) = Sum_{j=1..n+1} A007318(n,j-1) * binomial(k,j).

T(n,k) = A324999(n,k) + T(n,k-n) = (A324999(n,k) - A325001(n,k)) / 2 = T(n,k-n) + A325001(n,k). - Robert A. Russell, Sep 28 2020

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

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

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

T(n,k-n) = A324999(n,k) - T(n,k) = (A324999(n,k) - A325001(n,k)) / 2 = T(n,k) - A325001(n,k). - Robert A. Russell, Oct 10 2020

Example

The array begins with T(1,1):
  1  3  6  10  15   21   28    36    45    55    66     78     91    105 ...
  1  4 10  20  35   56   84   120   165   220   286    364    455    560 ...
  1  5 15  35  70  126  210   330   495   715  1001   1365   1820   2380 ...
  1  6 21  56 126  252  462   792  1287  2002  3003   4368   6188   8568 ...
  1  7 28  84 210  462  924  1716  3003  5005  8008  12376  18564  27132 ...
  1  8 36 120 330  792 1716  3432  6435 11440 19448  31824  50388  77520 ...
  1  9 45 165 495 1287 3003  6435 12870 24310 43758  75582 125970 203490 ...
  1 10 55 220 715 2002 5005 11440 24310 48620 92378 167960 293930 497420 ...
  ...
For T(1,2) = 3, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses two colors. For T(2,2)=4, the triangle may have 0, 1, 2, or 3 edges of one color.

Mathematica

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

Crossrefs

Cf. A324999 (oriented), A325001 (achiral).

Unoriented: A007318(n,k-1) (exactly k colors), A327084 (edges, ridges), A337884 (faces, peaks), A325005 (orthotope facets, orthoplex vertices), A325013 (orthoplex facets, orthotope vertices).

Chiral: A327085 (edges, ridges), A337885 (faces, peaks), A325006 (orthotope facets, orthoplex vertices), A325014 (orthoplex facets, orthotope vertices).

Cf. A104712 (same sequence for a triangle; same sequence apart from offset).

Rows 1-4 are A000217, A000292, A000332(n+3), A000389(n+4). - Robert A. Russell, Sep 28 2020

Sequence in context: A286158 A185915 A086270 * A104712 A122177 A255874

Adjacent sequences: A324997 A324998 A324999 * A325001 A325002 A325003

Keyword

nonn,tabl,easy

Author

Robert A. Russell, Mar 23 2019

Status

approved