Order-chain polytopes - Hibi, Takayuki (Ars mathematica contemporanea, 2019)
Given two families ?$X$? and ?$Y$? of integral polytopes with nice combinatorial and algebraic properties, a natural way to generate a new class of polytopes is to take the intersection ?$\mathcal P = \mathcal P_1 \cap \mathcal P_2$?, where ?$\mathcal P_1 \in X,\ \mathcal P_2 \in Y$?. Two basic questions then arise: 1) when ?$\mathcal P$? is integral and 2) whether ?$\mathcal P$? inherits the "old type" from ?$\mathcal P_1,\ \mathcal P_2$? or has a "new type", that is, whether ?$\mathcal P$? is unimodularly equivalent to a polytope in ?$X \cup Y$? or not. In this paper, we focus on the families of order polytopes and chain polytopes. Following the above framework, we create a new class of polytopes which are named order-chain polytopes. When studying their volumes, we discover a natural relation with Ehrenborg and Mahajan's results on maximizing descent statistics.
Keywords: delno urejena množica; order-chain polytope; poset; unimodular equivalence; unimodularna ekvivalenca; urejenostno-verižni politop; volume