A325003 Triangle read by rows: T(n,k) is the number of achiral colorings of the facets (or vertices) of a regular n-dimensional simplex using exactly k colors.

1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 10, 10, 5, 0, 1, 6, 15, 20, 15, 6, 0, 1, 7, 21, 35, 35, 21, 7, 0, 1, 8, 28, 56, 70, 56, 28, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 0

Offset

1,4

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. An achiral coloring is the same as its reflection. For k <= n all the colorings are achiral.

The final zero in each row indicates no achiral colorings when each facet has a different color.

Links

Robert A. Russell, Table of n, a(n) for n = 1..1325

Formula

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

T(n,k) = A325002(n,k) - 2*[k==n+1] = 2*A007318(n,k-1) - A325002(n,k).

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

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

Example

Triangle begins with T(1,1):
1  0
1  2   0
1  3   3   0
1  4   6   4    0
1  5  10  10    5    0
1  6  15  20   15    6    0
1  7  21  35   35   21    7    0
1  8  28  56   70   56   28    8    0
1  9  36  84  126  126   84   36    9    0
1 10  45 120  210  252  210  120   45   10    0
1 11  55 165  330  462  462  330  165   55   11    0
1 12  66 220  495  792  924  792  495  220   66   12   0
1 13  78 286  715 1287 1716 1716 1287  715  286   78  13   0
1 14  91 364 1001 2002 3003 3432 3003 2002 1001  364  91  14  0
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 0
For T(3,2)=3, the tetrahedron may have one, two, or three faces of one color.

Mathematica

Table[Binomial[n, k-1] - Boole[k==n+1], {n, 1, 15}, {k, 1, n+1}] \\ Flatten

Crossrefs

Cf. A325002 (oriented), A007318(n,k-1) (unoriented), A325001 (up to k colors).

Other n-dimensional polytopes: A325011 (orthotope), A325019 (orthoplex).

Cf. A198321.

Sequence in context: A155584 A139600 A198321 * A166278 A365515 A316269

Adjacent sequences: A325000 A325001 A325002 * A325004 A325005 A325006

Keyword

nonn,tabf

Author

Robert A. Russell, Mar 23 2019

Status

approved