2 problems

Let ${\overline w}\in{\mathbb Z}^r$, let $\alpha,\beta\geq 1$ satisfy $0\leq w_i<\alpha+\beta$ and $\gcd(\alpha,\beta)=1$, and let $a_i=\sharp\{j\mid w_j=i\}$. Let $\alpha'$ be the…

Let $p$ be a positive integer, and let $\mathcal{H}^{p}_{\hat a,\hat b}$ be the set of paths with boundary parameters $\hat a,\hat b$ and renormalised weight $\widehat{wt}(h)$. Let…