2016/02/01 by Ilia Nekrasov, Gaiane Panina, Nekrasov, Ilia +3
Mathematics · #51M20 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.AT #math.CO #math.MG #msc:51M20
paper · pdf · doi:10.48550/arxiv.1602.00471
arXiv admin note: text overlap with arXiv:1505.00352
arxiv created 2016/02/01 · arxiv updated 2016/02/02
The face poset of the permutohedron realizes the combinatorics of linearly ordered partitions of the set [n]=\1,...,n\. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set [n+1]. The cyclopermutohedron was introduced by the third author by motivations coming from configuration spaces of polygonal linkages. In the paper we prove two facts: (1) the volume of the cyclopermutohedron equals zero, and (2) the homology groups Hk for k=0,...,n-2 of the face poset of the cyclopermutohedron are non-zero free abelian groups. We also present a short formula for their ranks.