OFFSET
1,2
COMMENTS
Let V be an n-vector of the numbers 1 to n in sequence and let W be an n-vector of a signed permutation of these numbers. Numbers in W may be either positive or negative. a(n) is the number of vectors W which allow the scalar product V*W = 0. We include identity as a permutation. We allow all combinations of positive and negative signs in W; this means 2^n combinations of signs.
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 one vertex of such a permutohedron and define a hyperplane through the origin by the normal vector pointing from this vertex to the origin, then a(n) is the number of vertices which lie on this hyperplane.
The group G(r, n) is the group of n X n matrices such that there is exactly one nonzero entry in each row and column, and this entry is an r-th root of unity. Let the matrix M be a member of G(2, n) (hyperoctahedral group), then a(n) is the number of M where V'*(M*V) = 0 holds. Each vertex of our Coxeter permutohedron C_n corresponds to one such M acting as M*V. V is here an n-vector [1, 2, ..., n]. V' means the transpose of V.
LINKS
Bert Dobbelaere, Python program
FORMULA
Trivially, a(n) <= n! * (2^n - 1). - Charles R Greathouse IV, Jan 07 2023
EXAMPLE
The cases of a(4) = 16 are:
[1, 2, 3, 4] * [-4,-1,-2, 3] = -4 - 2 - 6 + 12 = 0.
[1, 2, 3, 4] * [-2,-3, 4,-1] = -2 - 6 + 12 - 4 = 0.
[1, 2, 3, 4] * [-3,-4, 1, 2] = -3 - 8 + 3 + 8 = 0.
[1, 2, 3, 4] * [-4,-3, 2, 1] = -4 - 6 + 6 + 4 = 0.
[1, 2, 3, 4] * [-2, 1,-4, 3] = -2 + 2 - 12 + 12 = 0.
[1, 2, 3, 4] * [-4, 3,-2, 1] = -4 + 6 - 6 + 4 = 0.
[1, 2, 3, 4] * [-2, 1, 4,-3] = -2 + 2 + 12 - 12 = 0.
[1, 2, 3, 4] * [-3, 4, 1,-2] = -3 + 8 + 3 - 8 = 0.
[1, 2, 3, 4] * [ 4, 1, 2,-3] = 4 + 2 + 6 - 12 = 0.
[1, 2, 3, 4] * [ 2, 3,-4, 1] = 2 + 6 - 12 + 4 = 0.
[1, 2, 3, 4] * [ 3, 4,-1,-2] = 3 + 8 - 3 - 8 = 0.
[1, 2, 3, 4] * [ 4, 3,-2,-1] = 4 + 6 - 6 - 4 = 0.
[1, 2, 3, 4] * [ 2,-1, 4,-3] = 2 - 2 + 12 - 12 = 0.
[1, 2, 3, 4] * [ 4,-3, 2,-1] = 4 - 6 + 6 - 4 = 0.
[1, 2, 3, 4] * [ 2,-1,-4, 3] = 2 - 2 - 12 + 12 = 0.
[1, 2, 3, 4] * [ 3,-4,-1, 2] = 3 - 8 - 3 + 8 = 0.
PROG
(PARI) a(n) = my(v=[1..n], nb=0); forperm(v, p, for (i=1, 2^(n-1)-1, my(w = vector(n, k, if (bittest(i, k-1), -p[k], p[k]))); if (v*w~ == 0, nb++); ); ); 2*nb; \\ Michel Marcus, Nov 30 2022
(PARI) a(n)=my(m=n-1, N=2^m-1, s); forperm(n, U, for(i=1, n, U[i]*=2*i); my(t=sum(i=1, n, U[i])/2); for(i=1, N, t+=U[valuation(i, 2)+1]*=-1; if(t==0, s++))); 2*s \\ Charles R Greathouse IV, Jan 07 2023
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Thomas Scheuerle, Nov 24 2022
EXTENSIONS
a(12)-a(20) from Bert Dobbelaere, Jan 26 2023
STATUS
approved