A325002 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets (or vertices) of a regular n-dimensional simplex using exactly k colors.

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

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 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. The only chiral pair occurs when k=n+1; for k <= n all the colorings are achiral.

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) = 2*A007318(n,k-1) - A325003(n,k) = [k==n+1] + A325003(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  2
1  2   2
1  3   3   2
1  4   6   4    2
1  5  10  10    5    2
1  6  15  20   15    6    2
1  7  21  35   35   21    7    2
1  8  28  56   70   56   28    8    2
1  9  36  84  126  126   84   36    9    2
1 10  45 120  210  252  210  120   45   10    2
1 11  55 165  330  462  462  330  165   55   11    2
1 12  66 220  495  792  924  792  495  220   66   12   2
1 13  78 286  715 1287 1716 1716 1287  715  286   78  13   2
1 14  91 364 1001 2002 3003 3432 3003 2002 1001  364  91  14  2
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 2
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. A007318(n,k-1) (unoriented), A325003 (achiral), A325001 (up to k colors).

Other n-dimensional polytopes: A325008 (orthotope), A325016 (orthoplex).

Sequence in context: A173306 A276430 A362325 * A182594 A201593 A029287

Adjacent sequences: A324999 A325000 A325001 * A325003 A325004 A325005

Keyword

nonn,tabf

Author

Robert A. Russell, Mar 23 2019

Status

approved