A proof of the Athanasiadis-Chapoton ordinal-sum conjecture

@misc{luo2026proof,
  title = {{A proof of the Athanasiadis-Chapoton ordinal-sum conjecture}},
  author = {Congyi Luo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28507},
  eprint = {2609.28507},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28507},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - A proof of the Athanasiadis-Chapoton ordinal-sum conjecture
AU  - Congyi Luo
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.28507
UR  - https://arxiv.org/abs/2609.28507
AB  - Given a finite preorder τ, we study how its order relation determines the h^*-polynomial of the associated polar preorder polytope. Let ℛ_τ^∨ be the convex hull of the negative standard basis vectors and the indicator vectors of nonempty order ideals, and write h_τ^*(t)=h^*(ℛ_τ^∨,t). Athanasiadis and Chapoton conjectured that this enumerative invariant takes ordinal sums to products: h_τ_1⊕τ_2^*(t)=h_τ_1^*(t) h_τ_2^*(t). We prove the conjecture for arbitrary finite preorders, including those with nontrivial equivalence classes. More precisely, we construct a positively homogeneous piecewise unimodular shear from the free sum ℛ_τ_1^∨⊕ℛ_τ_2^∨ onto ℛ_τ_1⊕τ_2^∨, giving lattice-point bijections at every nonnegative real dilation scale. The key is a formula for the Minkowski functional in terms of the least nonnegative order-reversing majorant: after the shear, the functional becomes the sum of the two factor functionals. Together with the Ehrhart product formula for free sums, this proves the conjecture and the corresponding product formula for normalized volume.
ER  -