Question

In an excerpt of an article by Bernd Sturmfels, I found:

Theorem 5.5. The tropical Grassmannian $G^{′}_{2,n}$ is a simplical complex known as the space of phylogenetic trees.... It is denoted by $T_n$ and is defined as follows. The vertex set consists of all unordered pairs $\left \{ A,B \right \}$ where $A$ and $B$ are disjoint subsets of $\left [ n \right ]:=\left \{ 1,2, ... , n \right \}$ having cardinality at least two, and $A \cup B=\left [ n \right ]$. Such pairs are called splits. The number of splits is $2^{n−1} − n − 1$. Two splits $\left \{ A,B \right \}$ and $\left \{ A^{'},B^{'} \right \}$ are connected by an edge in the simplicial complex $T_n$ if and only if

(5) $A \sqsubseteq {A}'$ or $A \sqsubseteq {B}'$ or $B \sqsubseteq {A}'$ or $B \sqsubseteq {B}'$.

We define $T_n$ as the largest simplicial complex having this edge graph.... In the language of algebraic combinatorics, $T_n$ is the flag complex of the compatibility graph specified by (5) on the set of all $2^{n−1} − n − 1$ splits.

Example 5.6. ($n = 6$) The two-dimensional simplicial complex $T_6$ has $25$ vertices, $105$ edges and $105$ triangles...

Question: Are $56, 490, 1260, 945$ the "face" numbers for $T_7$?

Answer 1

You have the right numbers. $\mathrm{Trop} \ G(2,n)$ is the space of phylogenetic trees studied in Billera-Holmes-Vogtman. It has $1 \times 3 \times 5 \times \cdots \times (2n-5)$ maximal faces and $2^{n-1} - n-1$ vertices, matching your $945$ and $56$.

The first several face numbers are $$\begin{matrix} 1 & & & & \\ 1 & 3 & & & \\ 1 & 10 & 15 & & \\ 1 & 25 & 105 & 105 & \\ 1 & 56 & 490 & 1260 & 945 \\ \end{matrix}$$

Write $f(n,k)$ for the number of $k$ faces in $\mathrm{Trop} \ G(2,n)$, we have $$f(n,k) = (k+1) f(n-1,k) + (n+k-2) f(n-1,k-1).$$ I am prone to off by one errors, so check this against the above table.

It is easy to get closed formulas for $f(n,r)$ or $f(n,n-r)$, for small fixed $r$, but there seems to be no simple expression for the middle terms. Billera-Holmes-Vogtman say that there is more discussion in Exercise 5.40 of Enumerative Combinatorics I by Stanley; I don't have my copy with me to check.