  We introduce the notion of a cut cellular surface (CCS), being a surface with
boundary, which is cut in a specified way to be represented in the plane, and
is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under
Pachner-like moves on the cellular structure, by counting colourings of the
1-cells with elements of a finite group, subject to a "flatness" condition for
each 2-cell. These invariants are also described in a TQFT setting, which is
not the same as the usual 2-dimensional TQFT framework. We study the properties
of functions which arise in this context, associated to the disk, the cylinder
and the pants surface, and derive general properties of these functions from
topology, including properties which come from invariance under the
Hatcher-Thurston moves on pants decompositions.
