n,k,m is a geometric object which generalizes
the positive Grassmannian (when n = k+m) and cyclic polytopes (when k=1). It was originally introduced in the context of scattering amplitudes. Of substantial interest are the tilings of the amplituhedron, which are analogous to triangulations of a polytope.
In \cite{karp2020decompositions}, it was 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
binom(n-2,k).
We prove this by showing that all positroid tilings of the hypersimplex
Δk+1,n have cardinality binom(n-2,k), then applying
T-duality. In addition, we give volume formulas for
Parke-Taylor polytopes and tree positroid polytopes
in terms of circular extensions of cyclic partial orders; and we prove
new variants of the
classical Parke-Taylor identities.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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