Second post about my summer research. This is a jump from the last one, but I need to clear my head. Here it goes.
After about two weeks of lots of reading, I've begun my independent inquiries. This will (and has to, to some extent) sound like pure brain-storming, incoherent babbling; but I need to keep track of where I want to get to.
We know (cite here) that the $h$-vector of a nestohedron $P_{\mathcal{B}}$ can be described in as the generating function of the descent number of its $\mathcal{B}$-trees:
$$h_{P_{\mathcal{B}}}(t) = \sum_T t^{\text{des}(T)}.$$
Given how there exists a bijection (cite here) between the vertices of $P_{\mathcal{B}}$, the $\mathcal{B}$-permutations and the $\mathcal{B}$-trees, it is clear that to understand the combinatorial structure of a nestohedra, we must understand the structure of its underlying building set.
In the case of graph-associahedra, I want to find results about their $h$- and $\gamma$-polynomials in constructive and intuitive ways. I've started working on finding that information about the graph-associahedra $P_{\mathcal{B}(\text{Cycle}_n)}$ of the cycle graph on $n$ vertices.
Let's suppose that we fully know what's the combinatorial structure of $P_{\mathcal{B}(\text{Path}_n)}$ (we do), and that we don't know anything about $P_{\mathcal{B}(\text{Cycle}_n)}$ (we do as well, but bare with me). If you consider the graph $\text{Path}_{(n-1)}$ with two added edges, from endpoints $1$ and $(n-1)$ to a new vertex $r$, you've essentially pictured $\text{Cycle}_n$ in your head. The idea of describing the graph-associahedra of the cycle graph in terms of the simpler path graph is cool. In fact, the idea of describing the graph-associahedra of any graph by `gluing' smaller, more understandable (or more understood) graphs is extremely attractive and logical to me. It makes sense to reuse simple constructs to provide descriptions of more complex ideas; but of course, it isn't easy. I don't even know if I'll be able to finish the example presented here, but it is a good warm-up exercise for this poor soul's first attempt at providing novel insight, no matter how insignificant.
I was feeling like finishing the post here, but i realize i haven't actually delved into the complicated stuff, and i want to reach the point where i can't go further, and observe then where's the weakness, where's the limitation.
We can define the set of $\mathcal{B}$-trees for $\text{Cycle}_n$ as follows: for each $r \in [n]$, construct the set of trees of $\text{Path}_{(n-1)}$ such that the label $i$ of every vertex of a tree $T$ is replaced by $i' = i + r \pmod{n}$. Join the root of $T$ to our newly defined root $r$, to get $T'$. Notice then that the forest of trees constructed from $\mathcal{B}(\text{Cycle}_n)$ has $n \cdot C_{(n-1)}$ trees ($C_n$ is the $n$th Catalan number; it is known that there are $C_n$ trees for the $n$-path).
So, to find $h_{P_{\mathcal{B}(\text{Cycle}_n)}}(t)$, we must count the descent numbers of the $n$ copies of the $\mathcal{B}$-forest of $\text{Path}_{(n-1)}$ with its labels shifted $r \in [n]$ modulo $n$. I hope you can see that the delicate part is the shifting of labels: if $n=5$ and in some tree from $\text{Path}_4$ we had a descent $4 \leftarrow 5$, when we adjoin the new root $r=1$, all indices are shifted and the descent becomes $5 \leftarrow 1$, which is not a descent anymore. Chaos! We must understand how does that shifting affect the descent number of the trees of $\text{Cycle}_n$. Here's where i'm at.
A small observation, to close this post. It may be obvious, but it has just occurred to me and i haven't come up with a proof from the top of my head: if you pick a tree $T$ from the forest of $\mathcal{B}(\text{Cycle}_n)$-trees, and exchange the root with its unique child, the resulting tree $T'$ is also a tree in the forest. If you apply the operation twice, you obviously go back to $T$. If closure were true, this would define an equivalence relation which halves the size of the forest. Postnikov et al. use a `similar' (this deserves triple quotes) operation to fully describe the graph-associahedra of the path graph. I may try to do something similar? But i realize that I'm not using what i know about the descents of the trees of the path graph itself! Back to work.
