Theorem 19.24

Theorem 19.24 (Exact depth-threshold profiles). Let $D_{k}^{\ge r}$ be the balanced words over $k$ bracket types whose stack height reaches at least $r\ge1$. Then $$q_{k,r}^{\ge}(N) =1+\sum_{h=0}^{N}k^h +\sum_{h=\max(0,2r-N)}^{r-1}k^h,$$ $$a_{k,r}^{\ge}(N)=\max(2r,N).$$ For the complementary balanced language $D_k^{<r}$ that stays below $r$, $$q_{k,r}^{<}(N) =1+\sum_{h=0}^{\min(N,r-1)}k^h, \qquad a_{k,r}^{<}(N)=\max(1,\min(N,r-1)).$$ The formulas remain unchanged when neutral letters are added.