The associahedron has edges of the form $a(bc)\rightarrow (ab)c.$ But I also want to include the possibility of swapping adjacent entries by doing operations like $a(bc) \rightarrow a(cb).$ I was wondering if there was some type of combinatorial structure that could help me. My main goal is to find an algorithm that can use these kinds of moves to swap around the $i$th and $j$th entries of some starting structure. For example, I could swap the first and third entries by doing $(ab)c \rightarrow (ba)c \rightarrow b(ac) \rightarrow b(ca) \rightarrow (bc)a \rightarrow (cb)a.$
It would be great to know any references or algorithms I should look at. My interest in this problem comes from thinking about symmetric monoidal categories. I suppose it relates to the permutahedron as well.
