 One-dimensional entropic chain force-extension relation

For $N$ independent links pointing left or right with length $a$, fixed signed extension $L$ gives $N_R=(N+L/a)/2$ and $N_L=(N-L/a)/2$. The multiplicity is $\left(\genfrac{}{}{0ex}{}{N}{N_L}\right)$ when these counts are nonnegative integers. Leading Stirling's approximation gives $\mathcal{F}=(k_{B}T/(2a))\ln (N_R/N_L)$ as the externally applied tension conjugate to $L$. Its small-extension limit is $k_{B}TL/(N{a}^2)$. At fixed tension, the exact one-link partition function is $2\cosh (\beta \mathcal{F}a)$ and the mean extension is $Na\tanh (\beta \mathcal{F}a)$, confirming the same inverse relation without using Stirling at the full-extension endpoint.