A325003 - OEIS

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

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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