On the topology of bi-cyclopermutohedra

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 $\{1,\cdots, n+1\}$ 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.