* Let $a(n)$ be [A309099][1] i.e. the number of partitions of $n$ avoiding the partition $(4,3,1)$.

* We say a partition $\alpha$ contains $\mu$ provided that one can delete rows and columns from (the Ferrers board of) $\alpha$ and then top/right justify to obtain $\mu$.  If this is not possible then we say $\alpha$ avoids $\mu$.  For example, the only partitions avoiding $(2,1)$ are those whose Ferrers boards are rectangles.

* Let
$$
b(n) = n + \sum\limits_{i=1}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\sum\limits_{j=1}^{n-2i}\left\lfloor\frac{i+j-1}{j+1}\right\rfloor
$$

I conjecture that $$b(n) = a(n).$$

Is there a way to prove it?

  [1]: https://oeis.org/A309099