A381339 Number of vector differences between two permutations of order n, up to multiplication by nonzero rational numbers and permutations of the components.

1, 1, 2, 3, 9, 28, 128, 539, 2651, 13000, 67466, 355381, 1926343, 10590537, 59234734, 335302599

Offset: 0

Author

Max Alekseyev, Feb 20 2025

Keywords

• nonn
• more
• hard

Comments

Nonzero difference vectors are associated with their images in the projective space, and in addition we do not distinguish vectors that can be permuted one into the other. In the affine space, their number (including zero vector) is given by A019589, implying that a(n) <= A019589(n). Nonzero difference vectors in the projective space are counted by A381243.

Examples

			For n = 3, there are A019589(3) = 5 difference vectors up to permutation of components: (-2, 0, 2), (-2, 1, 1), (-1, -1, 2), (-1, 0, 1), and (0, 0, 0). However, (-2, 0, 2) and (-1, 0, 1) are the same up to a factor 2, and (-2, 1, 1) and (-1, -1, 2) are the same up to negation and reversing the order. Hence, a(3) = 3.
		

Crossrefs

Cf. A019589, A175176, A362968, A381243, A381244.