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Fix a natural number $n$, and let $O_n$ be the set of ordered partitions of $n$. For example $O_3=\{1+1+1,1+2,2+1,3\}$ which can also be written as $\{1|2|3,1|23,12|3,123\}$.

We can define two binary operations $\star$, $\bullet$ on $O_n$. It is easiest to do this by example. Consider $O_6$. First we work with $\star$:

$$1|234|56\star1|2|3456=1|2|34|56$$ $$(1+3+2)\star(1+1+4)=(1+1+2+2)$$

The $\star$ sum of two ordered partitions has a $|$ between $m$ and $m+1$ if and only if at least one of the ordered partitions themselves has a $|$ between $m$ and $m+1$.

Similarly, for $\bullet$:

$$1|234|56\bullet1|2|3456=12|34|56$$ $$(1+3+2)\bullet(1+1+4)=(2+2+2)$$

Here,the $\bullet$ sum of two ordered partitions has a $|$ between $m$ and $m+1$ if and only if exactly one of the ordered partitions themselves has a $|$ between $m$ and $m+1$.

Both of these operations give $O_n$ the structure of a commutative monoid (in fact $\bullet$ has inverses, so $O_n$ is an abelian group under $\bullet$). Have these monoid structures on ordered partitions been studied at all? Can anybody point me to a reference? Also, is there a natural way to place a monoid structure on the set $P_n$ of (unordered) partitions of a natural number $n$?

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Have I misunderstood something, or is it the case that $\star$ is essentially the bitwise-or operation on $n-1$-bit numbers, and that $\bullet$ is essentially the bitwise-exclusive-or operation? If so, then the answer is yes, and the binary number representation seems to me to be much more perspicuous. –  MJD Mar 5 at 19:54

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