The Type A component symmetric-matrix formula

Let $\mathcal{S}$ be the set of symmetric binary matrices with no row sum greater than one. For $s\in\mathcal{S}$, let $A(s)$ be the set of upper-triangular positions containing a one, define $F(s)$ as in the source, call $s$ admissible when no two links satisfy $i<i'<j<j'$, and let $\operatorname{Adm}(b)$ be the admissible matrices mapped to the binary string $b$. For a Type A diagram $D(b)$, Type A component formula.

$$\Psi_{D(b)}=\sum_{s\in\operatorname{Adm}(b)}F(s).$$

The formula is presented inside a conjecture environment and gives an explicit combinatorial expression for selected ground-state components; no proof or resolution is supplied.