login
Coordination sequence for Leech lattice.
1

%I #32 Jun 14 2024 22:31:10

%S 1,196560,39462040800

%N Coordination sequence for Leech lattice.

%C Coordination sequence for graph G having a vertex for each point of the Leech lattice, with each vertex joined by an edge to its 196560 nearest neighbors.

%C No formula or recurrence is presently known (see page xxxvi of the sphere packing book).

%D J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. 181.

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993; see page xxxvi and Tables 4.13 and 4.14.

%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).

%e The graph G has degree 196560, so a(1)=196560.

%e Let L(i) denote the set of vectors in the Leech lattice having norm 2i, and let c(i) = |L(i)| = A008408(i). The first few sets L(i) are described on page 181 of the ATLAS and in Table 4.13 of the sphere packing book.

%e The vertices at edge-distance 2 from the vertex 0^24 consist of the sets L(3), L(4), L(5), L(6), and the doubles of L(2), so a(2) = c(2)+c(3)+...+c(6) = 39462040800.

%e To prove this, note that the group Aut(G) = Co_0 acts transitively on each of the sets L(2), L(3), L(4), L(5), and L(7) (cf. A323273). It is easy to find one representative from each set L(3), L(4), L(5) among the sums of pairs of minimal vectors. There are two orbits on L(6) and again it is easy to find a representative from each orbit, for example

%e [3, 3, 3, 3, 3, 3, 3, 3, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]

%e and

%e [2, 2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2].

%e Among the sums of pairs of minimal vectors there are no vectors from L(7), so there is no contribution from c(7).

%e Finally, the only possibility for a vector of norm 16 is the double of a minimal vector, and there are c(2) = 196560 of these, all in one obit under the group.

%e So a(2) = c(3)+c(4)+c(5)+c(6)+c(2) = 39462040800.

%Y Cf. A007900, A008340, A008408, A323273.

%K nonn,more,bref

%O 0,2

%A _N. J. A. Sloane_, Jan 12 2019