A358655 a(n) is the number of distinct scalar products which can be formed by pairs of signed permutations (V, W) of [n].

1, 2, 7, 24, 61, 111, 183, 281, 409, 571, 771, 1013, 1301, 1639, 2031, 2481, 2993, 3571, 4219, 4941, 5741, 6623, 7591, 8649, 9801, 11051, 12403, 13861, 15429, 17111, 18911, 20833, 22881, 25059, 27371, 29821, 32413, 35151, 38039, 41081, 44281, 47643

Offset

0,2

Comments

Let V be an n-vector of the numbers 1 to n in order and let W be an n-vector of any signed permutation of these numbers. Numbers in W may be either positive or negative. a(n) is the number of different values for the scalar product V*W for all possible W. We allow all combinations of positive and negative signs in W.

Another interpretation of this sequence: A signed permutohedron is also called the Coxeter permutohedron of the family C_n and has A000165(n) vertices. If we choose a vector of one vertex of such a permutohedron to the origin, and cut this permutohedron in slices by hyperplanes which are orthogonal to this vector such that in each slice lies at least one vertex of this permutohedron, then a(n) is the count of such slices obtained by this process.

a(n) is odd if the plane through the origin is occupied by vertices, this means A358629(n) > 0.

For n > 3 all possible planes are occupied by vertices and thus a nice formula ( see formula section ) exists.

The permutohedra for small n are:

n = 2 Octagon.

n = 3 Truncated cuboctahedron.

n = 4 Omnitruncated tesseract.

n = 5 Omnitruncated 5-cube.

Links

Table of n, a(n) for n=0..41.

Thomas Scheuerle, The permutohedron for a(3) a truncated cuboctahedron.

Thomas Scheuerle, The truncated cuboctahedron of a(3) with 24 planes intersecting in at least one vertex.

Thomas Scheuerle, The octagon of a(2) with 7 planes intersecting in at least one vertex.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = (2*n^3 + 3*n^2 + n + 3)/3 = A188475(n), for n > 3 (because valid if A000165(n)/2 > A188475(n)).

From Stefano Spezia, Nov 28 2022: (Start)

G.f.: (1 - 2*x + 5*x^2 + 4*x^3 - 15*x^5 + 16*x^6 - 5*x^7)/(1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 7. (End)

Example

a(2) = 7
Columns in the table below:
A: Result of the scalar product.
B: Count of combinations for this result.
C: An example.
    A  B      C
    5  1  [ 2,  1]*[ 2, 1]
    4  1  [ 1,  2]*[ 2, 1]
    3  1  [ 2, -1]*[ 2, 1]
    0  2  [ 1, -2]*[ 2, 1]
   -3  1  [-2,  1]*[ 2, 1]
   -4  1  [-1, -2]*[ 2, 1]
   -5  1  [-2, -1]*[ 2, 1]
We have 7 rows. The sum over B is A000165(2).
For a(2) all vectors C are part of the vertices of an octagon.

Crossrefs

Cf. A000165, A188475, A358629.

Sequence in context: A112089 A075062 A022497 * A258341 A054128 A082047

Adjacent sequences: A358652 A358653 A358654 * A358656 A358657 A358658

Keyword

nonn,easy

Author

Thomas Scheuerle, Nov 25 2022

Status

approved