Arithmetic of marked poset polytopes, monotone triangle reciprocity,and partial colorings

Arithmetic of marked poset polytopes, monotone triangle reciprocity,and partial colorings

Katharina Jochemko, Raman Sanyal – 2012

Focus Area 1: High-complexity Geometry
For a poset P, a subposet A, and an order preserving map F from A into the real numbers, the marked order polytope parametrizes the order preserving extensions of F to P. We show that the function counting integral-valued extensions is a piecewise polynomial in F and we prove a reciprocity statement in terms of order-reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler (2011) and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty (2007).

Title

Arithmetic of marked poset polytopes, monotone triangle reciprocity,and partial colorings