Counting faces of graphical zonotopes

Abstract

It is a classical fact that the number of vertices of the graphical zonotope ZΓ is equal to the number of acyclic orientations of a graph Γ. We show that the f -polynomial of ZΓ is obtained as the principal specialization of the q-analog of the chromatic symmetric function of Γ.

Weighted quasisymmetric enumerator for generalized permutohedra

“…The main argument in [12] of the proof of Theorem 4.4 for graphical zonotopes was based on the Humpert and Martin cancelation-free formula for the antipode of the chromatic Hopf algebra of graphs [14].…”

“…On the other hand, when a class of generalized permutohedra is specified we obtain the well known combinatorial enumerators. The case of graphical zonotopes Q = Z Γ is studied in [12]. The quasisymmetric function F q (Z Γ ) is a q-refinement of the Stanley chromatic symmetric function X Γ of graphs…”

“…The permutohedron P e n−1 = P Bn is realized as the nestohedron corresponding to the family B n of all subsets of [n]. Since rk Bn (L) = n − |L| for any chain L of subsets of [n] we have by(12) thatF q (P e n−1 ) = L q n−|L| M type(L) = α|=n n α q n−k(α) M α .Consequently by Theorem 4.4 we derive the well known factf (P e n−1 , q) = α|=n n α q n−k(α) . For the building set B = {{1}, .…”

“…Note that for connected B the contraction B/I remains connected for each I ⊂ [n] and rk(B/I) = n − |I| − 1. If we rearrange the sum in the expansion(12) according to predecessors of the maximal elements in chains we obtainF q (B) = I [n] q n−|I|−1 L I q rk B| I (L I ) M (type(L I ),n−|I|) , where the last sum is over all chains L I of I. This leads, by repeated application of equation (12) to the needed identity.…”

A brief survey on lattice zonotopes

“…Remark 6.5. Grujić [18] has shown that the f -polynomial of Z G , which encodes the number of faces of Z G in each dimension, can be obtained as the principal specialization of the q-analogue of the chromatic symmetric function of G.…”