This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A358629 #43 Jun 18 2023 13:46:14 %S A358629 0,2,0,16,48,558,4444,62246,692598,11722730,196824592,3896202680, %T A358629 86626174698,2018770217402,51681142218502,1418482891697258, %U A358629 41404316055037624,1304323691188387488,43501661519771535260,1538705372277647632786 %N A358629 a(n) is the number of signed permutations W of V = (1, 2, ..., n) such that the dot product V*W = 0. %C A358629 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. %C A358629 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. %C A358629 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. %H A358629 Bert Dobbelaere, <a href="/A358629/a358629.py.txt">Python program</a> %F A358629 Trivially, a(n) <= n! * (2^n - 1). - _Charles R Greathouse IV_, Jan 07 2023 %e A358629 The cases of a(4) = 16 are: %e A358629 [1, 2, 3, 4] * [-4,-1,-2, 3] = -4 - 2 - 6 + 12 = 0. %e A358629 [1, 2, 3, 4] * [-2,-3, 4,-1] = -2 - 6 + 12 - 4 = 0. %e A358629 [1, 2, 3, 4] * [-3,-4, 1, 2] = -3 - 8 + 3 + 8 = 0. %e A358629 [1, 2, 3, 4] * [-4,-3, 2, 1] = -4 - 6 + 6 + 4 = 0. %e A358629 [1, 2, 3, 4] * [-2, 1,-4, 3] = -2 + 2 - 12 + 12 = 0. %e A358629 [1, 2, 3, 4] * [-4, 3,-2, 1] = -4 + 6 - 6 + 4 = 0. %e A358629 [1, 2, 3, 4] * [-2, 1, 4,-3] = -2 + 2 + 12 - 12 = 0. %e A358629 [1, 2, 3, 4] * [-3, 4, 1,-2] = -3 + 8 + 3 - 8 = 0. %e A358629 [1, 2, 3, 4] * [ 4, 1, 2,-3] = 4 + 2 + 6 - 12 = 0. %e A358629 [1, 2, 3, 4] * [ 2, 3,-4, 1] = 2 + 6 - 12 + 4 = 0. %e A358629 [1, 2, 3, 4] * [ 3, 4,-1,-2] = 3 + 8 - 3 - 8 = 0. %e A358629 [1, 2, 3, 4] * [ 4, 3,-2,-1] = 4 + 6 - 6 - 4 = 0. %e A358629 [1, 2, 3, 4] * [ 2,-1, 4,-3] = 2 - 2 + 12 - 12 = 0. %e A358629 [1, 2, 3, 4] * [ 4,-3, 2,-1] = 4 - 6 + 6 - 4 = 0. %e A358629 [1, 2, 3, 4] * [ 2,-1,-4, 3] = 2 - 2 - 12 + 12 = 0. %e A358629 [1, 2, 3, 4] * [ 3,-4,-1, 2] = 3 - 8 - 3 + 8 = 0. %o A358629 (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 %o A358629 (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 %Y A358629 Cf. A000165, A358655. %K A358629 nonn,more %O A358629 1,2 %A A358629 _Thomas Scheuerle_, Nov 24 2022 %E A358629 a(12)-a(20) from _Bert Dobbelaere_, Jan 26 2023