4 Redid with LaTeX

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

Theorem 5.5. The tropical Grassmannian G′_2,n $G^{′}_{2,n}$ is a simplical complex known as the space of phylogenetic trees.... It is denoted by T_n $T_n$ and is defined as follows. The vertex set consists of all unordered pairs $\left \{ A,B} A,B \right \}$ where A $A$ and B $B$ are disjoint subsets of $\left [ n] := n \right ]:=\left \{ 1, 21,2, . .. .. , n } \right \}$ having cardinality at least two, and $A B = \cup B=\left [ n].n \right ]$. Such pairs are called splits. The number of splits is 2^(n−1) $2^{n−1} − n − 11$. Two splits $\left \{ A,B} A,B \right \}$ and $\left \{ A′,B′} A^{'},B^{'} \right \}$ are connected by an edge in the simplicial complex Tn $T_n$ if and only if

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

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

Example 5.6. (n $n = 66$) The two-dimensional simplicial complex T_6 $T_6$ has 25$25$ vertices, 105 $105$ edges and 105 $105$ triangles...

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

3 added 3 characters in body

In an extract 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 {A,B} where A and B are disjoint
subsets of [n] := {1, 2, . . . , n} having cardinality at least two, and A ∪ B = [n].
Such pairs are called splits. The number of splits is 2^(n−1) − n − 1. Two splits {A,B} and
{A′,B′} are connected by an edge in the simplicial complex Tn if and only if
(5) A ⊆ A′ or A ⊆ B′ or B ⊆ A′ or B ⊆ 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 2n−1 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?

2 Corrected formula as noted by commentator; added 2 characters in body

In an extract 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 {A,B} where A and B are disjoint
subsets of [n] := {1, 2, . . . , n} having cardinality at least two, and A ∪ B = [n].
Such pairs are called splits. The number of splits is 2n−1 2^(n−1) − n − 1. Two splits {A,B} and
{A′,B′} are connected by an edge in the simplicial complex Tn if and only if
(5) A ⊆ A′ or A ⊆ B′ or B ⊆ A′ or B ⊆ 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 2n−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?

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