  We develop a method for generating degree-of-freedom maps for arbitrary order
finite element spaces for any cell shape. The approach is based on the
composition of permutations and transformations by cell sub-entity. Current
approaches to generating degree-of-freedom maps for arbitrary order problems
typically rely on a consistent orientation of cell entities that permits the
definition of a common local coordinate system on shared edges and faces.
However, while orientation of a mesh is straightforward for simplex cells and
is a local operation, it is not a strictly local operation for quadrilateral
cells and in the case of hexahedral cells not all meshes are orientable. The
permutation and transformation approach is developed for a range of element
types, including Lagrange, and divergence- and curl-conforming elements, and
for a range of cell shapes. The approach is local and can be applied to cells
of any shape, including general polytopes and meshes with mixed cell types. A
number of examples are presented and the developed approach has been
implemented in an open-source finite element library.
