OFFSET
1,3
LINKS
A. Knutson, T. Tao, and C. Woodward, The honeycomb model of GL_n(C) tensor products II, J. Amer Math. Soc. 17(2004), 19-48.
EXAMPLE
A rigid honeycomb is extreme if it belongs to an extreme ray in the convex cone of honeycombs. Two extreme rigid honeycombs are equivalent if their exit rays have proportional multiplicities. The weight of such a honeycomb is one third the sum of the exit multiplicities, provided that all the multiplicities are positive integers with no common factors. There is exactly one equivalence class for weights 1 and 2, so a(1)=a(2)=1. Up to rotations and symmetries, there is also one equivalence class of weight 3, but we count the rotated/symmetric honeycombs are not equivalent, so we have a(3)=6. (The exit multiplicities of three of these are, for instance, 1,2|1,1,1|1,1,1, 2,1|1,1,1|1,1,1, and 1,1,1|2,1|1,1,1.) For n=4, there are just eight equivalence classes up to rotations and symmetries. We get a(4)=2+6*7 because one of these is equivalent to its rotations; it has exit multiplicities 2,1,1|2,1,1|2,1,1 or 1,1,2|1,1,2|1,1,2. Similarly, for n=5 there are two equivalence class that are rotationally symmetric with exit multiplicities 2,2,1|2,2,1|2,2,1 and 1,2,2|1,2,2|1,2,2 so a(5)=2 (mod 6). No precise count for a(6) but certainly a(6)>1000 and a(6)=0 (mod 6). Similarly,, a(7)=2 (mod 6), a(8)=2 (mod 6), and a(3n)=0 (mod 6). Note however that, for n=8, there are 8 (rather than 2) equivalence classes that have rotational symmetry.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Hari Bercovici, Jul 30 2019
STATUS
approved