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There is a concept of the so-called RP-compositions of an integer discussed by K. Q. Ji and H. S. Wilf in Extreme palindromes. They proved the following result too:

Theorem. The number of RP-compositions of $n$ is equal to the number of partitions of $n$ into powers of 2 (“binary partitions”).

I would like to ask:

QUESTION. Is there an object similar to RP-compositions which is equinumerous with partitions into powers of $3$ ("$3$-ary partitions")?

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  • $\begingroup$ Couldn't one just replace words, viewed in this way as labelled 2-regular graphs, by labelled 3-regular graphs, and require recursive symmetry under the obvious notion of rotating the 3 arms? $\endgroup$
    – LSpice
    Nov 19, 2021 at 23:21
  • $\begingroup$ The only minor complication is that RP orders the exceptional element of an odd-length sequence to the middle. The RP3 compositions can be defined recursively as the empty composition, $x+c+c+c$ where $x$ is a positive integer and $c$ is an RP3 composition, or $c+c+c$ as the degenerate case corresponding to $x=0$. $\endgroup$
    – Peter Taylor
    Nov 19, 2021 at 23:37

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