  The Z_k parafermionic conformal field theories, despite the relative
complexity of their modes algebra, offer the simplest context for the study of
the bases of states and their different combinatorial representations. Three
bases are known. The classic one is given by strings of the fundamental
parafermionic operators whose sequences of modes are in correspondence with
restricted partitions with parts at distance k-1 differing at least by 2.
Another basis is expressed in terms of the ordered modes of the k-1 different
parafermionic fields, which are in correspondence with the so-called multiple
partitions. Both types of partitions have a natural (Bressoud) path
representation. Finally, a third basis, formulated in terms of different paths,
is inherited from the solution of the restricted solid-on-solid model of
Andrews-Baxter-Forrester. The aim of this work is to review, in a unified and
pedagogical exposition, these four different combinatorial representations of
the states of the Z_k parafermionic models.
  The first part of this article presents the different paths and partitions
and their bijective relations; it is purely combinatorial, self-contained and
elementary; it can be read independently of the conformal-field-theory
applications. The second part links this combinatorial analysis with the bases
of states of the Z_k parafermionic theories. With the prototypical example of
the parafermionic models worked out in detail, this analysis contributes to fix
some foundations for the combinatorial study of more complicated theories.
Indeed, as we briefly indicate in ending, generalized versions of both the
Bressoud and the Andrews-Baxter-Forrester paths emerge naturally in the
description of the minimal models.
