A381244 -id:A381244 - cp's OEIS frontend

A381243 Number of hyperplanes defined by the nonzero differences of two permutations of order n.

0, 0, 1, 6, 85, 1370, 30481, 778610, 24409645, 881325366, 36635553601, 1713454403210, 89415912126223, 5143372266050837, 323667807885619744, 22112062644980805684

Offset: 0

Max Alekseyev, Feb 17 2025

Each of A175176(n) - 1 nonzero differences between two permutations (viewed as vectors) defines a hyperplane in the n-dimensional space. a(n) gives the number of pairwise distinct hyperplanes among them.

Cf. A019589, A175176, A362968, A381244, A381339.

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

Original entry on oeis.org

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

Offset: 0

Max Alekseyev, Feb 20 2025

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.

			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.

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

A384035 Number of vector differences between two permutations of order n, up to multiplication by positive rational numbers and permutations of the components.

Original entry on oeis.org

1, 1, 2, 4, 13, 49, 228, 1034, 5133, 25710, 133872, 708976, 3846150, 21170077, 118429072, 670537495

Offset: 0

Max Alekseyev, May 17 2025

			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. Hence, a(3) = 4.

Cf. A019589, A175176, A362968, A381243, A381244, A381339.

cp's OEIS Frontend

A381243 Number of hyperplanes defined by the nonzero differences of two permutations of order n.

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

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A384035 Number of vector differences between two permutations of order n, up to multiplication by positive rational numbers and permutations of the components.

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