The Magic Number Conjecture for the $m=2$ amplituhedron and Parke-Taylor identities

The amplituhedron An,k,m is a geometric object introduced in the context of scattering amplitudes in N=4 super Yang Mills. It generalizes the positive Grassmannian (when $n=k+m$), cyclic polytopes (when $k=1$), and the bounded complex of the cyclic hyperplane arrangement (when $m=1$). Of substantial interest are the tilings of the amplituhedron, which are analogous to triangulations of a polytope. Karp, Williams and Zhang (2020) observed that the known tilings of An,k,2 have cardinality n-2 k and the known tilings of An,k,4 have cardinality the Narayana number 1/n-3n-3 k+1n-3 k; generalizing these observations, they conjectured that for even m the tilings of An, k,m have cardinality the MacMahon number, the number of plane partitions which fit inside a k × (n-k-m) × m/2 box. We refer to this prediction as the `Magic Number Conjecture'. In this paper we prove the Magic Number Conjecture for the $m=2$ amplituhedron: that is, we show that each tiling of An,k,2 has cardinality n-2 k. We prove this by showing that all positroid tilings of the hypersimplex Δk+1,n have cardinality n-2 k, then applying T-duality. In addition, we give combinatorial necessary conditions for tiles to form a tiling of An,k,2; we give volume formulas for Parke-Taylor polytopes and certain positroid polytopes in terms of circular extensions of cyclic partial orders; and we prove new variants of the classical Parke-Taylor identities.