A270950 Number of distinct cardinalities of orbits of lattice points under the automorphism group of the n-dimensional integer lattice.

0, 1, 2, 5, 9, 12, 20, 29, 40, 53, 76, 99, 132, 172, 216, 270, 341, 424, 532, 660, 810, 983, 1210, 1446, 1750, 2111, 2508, 2975, 3569, 4197, 4948, 5807, 6817, 7963, 9351, 10863, 12604, 14598, 16892, 19439, 22472, 25780, 29588, 33892, 38800, 44206, 50463, 57297, 65086, 73919, 83842, 94510

Offset: 0

Author

Philippe A.J.G. Chevalier, Mar 26 2016

Keywords

• nonn
• hard

Comments

A finite number of orbits partition hypercubic shells of infinity norm s in the n-dimensional integer lattice. The number of orbits is given by C(n+s-1,s). The number of distinct cardinalities of the orbits of lattice points under the automorphism group of the n-dimensional integer lattice is found under the condition that n <= s.

A new connection was discovered using the partition of the dimension 'n'. These partitions create a base set of cardinalities. Each of these cardinalities can be subjected to the process of prime factorization. The prime factorization yields the exponents of the primes that form lattice points in a new integer lattice of dimension 'n'. These lattice points become elements of a set A. The unique summands of a specific partition of 'n' give the multipliers of the base vector (1,0^n) that need to be subtracted from the specific partition representative element of set A. The cardinality of the set A increases until all the specific partitions of 'n' have been processed. This augmented set A* has the correct cardinality. This method is much faster than the brute force technique. - Philippe A.J.G. Chevalier, Jun 24 2022

Examples

			For n=0 the a(0)=0.
For n=3 we have the following distinct cardinalities of the orbits 6, 8, 12, 24, 48 and thus a(3)=5.
For n=4 we have the distinct cardinalities of the orbits 8, 16, 24, 32, 48, 64, 96, 192, 384 and thus a(4)=9.
For n=5 we have the distinct cardinalities of the orbits 10, 32, 40, 160, 240, 320, 480, 640, 960, 1920, 3840 and thus a(5)=12.
		

Links

• Philippe A.J.G. Chevalier, Table of n, a(n) for n = 0..54

Crossrefs

Cf. A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287.

Extensions

a(17) corrected and a(18)-a(51) from Philippe A.J.G. Chevalier, Jun 24 2022