Rank formula for the first homology of token graphs of stars

Let $S_m$ be the star graph on $m+1$ vertices, and let $\tok{S_m}{n}$ denote its token graph. Write $A_1(-)$ for the first homology group. The graph $\tok{S_m}{n}$ is known to have free first homology. Rank formula for star token graphs. The free group $A_1(\tok{S_m}{n})$ has rank

$$(n-1)\binom{m}{n}-\binom{m}{n-1}+1.$$

This formula was checked computationally for $m\leq 11$ and $n\leq 7$, but the source does not provide a proof in general.