# Main Theorem

#### Main Theorem

The main result of our research lies in the following theorem:

Denoting $$q_\ell = n/$$gcd$$(n,2\ell+1)$$, we have: $VR(P_n;r)\simeq \begin{cases} \bigvee^{q_\ell-1}S^{2\ell}&\mbox{ when }s_{n,\ell} < r \le t_{n,\ell}\\ S^{2\ell+1}&\mbox{ when }t_{n,\ell} < r\le s_{n,\ell+1}\end{cases} \mbox{ for some } \ell \in \mathbb{N}$ and all of the summands in question are persistent.

As an example of the theorem, we can find the barcodes for $$P_{15}$$: