  The states in the irreducible modules of the minimal models can be
represented by infinite lattice paths arising from consideration of the
corresponding RSOS statistical models. For the M(p,2p+1) models, a completely
different path representation has been found recently, this one on a
half-integer lattice; it has no known underlying statistical-model
interpretation. The correctness of this alternative representation has not yet
been demonstrated, even at the level of the generating functions, since the
resulting fermionic characters differ from the known ones. This gap is filled
here, with the presentation of two versions of a bijection between the two path
representations of the M(p,2p+1) states.
  In addition, a half-lattice path representation for the M(p+1,2p+1) models is
stated, and other generalisations suggested.
