We analyze the existing encodings from MaxCSP into Partial MaxSAT, and report on a number of new insights that we have gained from our analysis, which can be summarized as follows: (i) the at-most-one (AMO) condition can be omitted in direct encodings from MaxCSP into Partial MaxSAT, and auxiliary variables are not needed; (ii) the sequential encoding of the cardinality constraint is, in fact, a reformulation of a regular encoding; (iii) the All Different constraint based on regular literals may be simplified; (iv) if we represent, in support encodings, the supporting values of a variable using intervals, then we can derive a genuine regular support encoding without exponential blowup; and (v) the Equal constraint admits a concise representation with regular signs.