Cyclical Partition Sequences

Consider the two sequences A_k and B_k,  k=1,2,3,... with the values
shown below

A:  1,2,3,4,6,9,11,15,19,25,31,41,49,61,75,91,110,134,157,189,...

B:  1,2,3,5,6,10,12,17,22,29,36,48,58,73,91,111,134,165,197,236,...

These two sequences are "duals" of each other in the sense that A_n
is the number of partitions of n into elements of B, and B_n is the
number of partitions of n into elements of A.

In general, given two sequences X and Y, we write X = P{Y} to indicate 
that X_n is the number of partitions of n into elements of Y.  Thus,
the above sequences satisfy the equations A=P{B} and B=P{A}.  [I've 
recently learned that these two sequences appear in Sloane's 
Encyclopedia of Integer Sequences.]

Does there exist a cycle of partition sequences X,Y,Z such that
X=P{Y}, Y=P{Z}, and Z=P{X}?

The answer is no.  A complete description of all possible periodic
sequences of partition sequences was emailed to me by Dan Ford.  His
analysis is summarized in Finding All Cycles of Partition Sequences