Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $P_i \to J_i \to K_{i+1}$. The projection is as follows: we index faces of $P_i$ with leveled planar binary trees with $i+1$ leaves, and we index faces of $K_{i+1}$ with planar binary trees with $i+1$ leaves; so $p_i$ is by disregarding levels.

However, one labelling of $J_i$'s faces that I know is by appropriately painted planar binary trees with $i$ leaves, not $i+1$ (as in Forcey https://arxiv.org/abs/0706.3226). So I am having a problem understanding the factorization above. How does one see painted trees with $i$ leaves as equivalence classes of leveled trees with $i+1$ leaves?

(In case $i=3$, $P_3$ and $J_3$ are both hexagons, so the equivalence classes are just with one element - there should be a bijection, but which bijection?)

  • $\begingroup$ Maybe use the level of the unique inner vertex under the rightmost leaf of the leveled tree to fix the painting height ? $\endgroup$
    – F. C.
    Oct 21, 2020 at 19:56


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