On the topology of bi-cyclopermutohedra

Abstract

Motivated by the work of Panina and her coauthors on cyclopermutohedron we study a poset whose elements correspond to equivalence classes of partitions of the set $$\{1,\dots , n+1\}$$ up to cyclic permutations and orientation reversion. This poset is the face poset of a regular CW complex which we call bi-cyclopermutohedron and denote it by $$\mathrm {QP}_{n+1}$$ . The complex $$\mathrm {QP}_{n+1}$$ contains subcomplexes homeomorphic to moduli space of certain planar polygons with $$n+1$$ sides up to isometries. In this article we find an optimal discrete Morse function on $$\mathrm {QP}_{n+1}$$ and use it to compute its homology with $${\mathbb {Z}}$$ as well as $${\mathbb {Z}}_2$$ coefficients.