The exact Carleson constant formula for postorder-rearranged dyadic collections

Let $N\in\mathbb{N}$ with $N\geq 2$ and let $0\leq \ell\leq N-2$. For the dyadic collection ${\mathcal E}^N_{\ell,0}$ and the postorder rearrangement operator $\tau_N$, write $\llbracket\tau_N({\mathcal E}^N_{\ell,0})\rrbracket$ for its Carleson constant, defined as the supremum of the normalized total lengths of subintervals in the collection. Exact Carleson constant formula. The supremum in the definition of $\llbracket\tau_N({\mathcal E}^N_{\ell,0})\rrbracket$ is attained for the interval $I_{1,0}$, and

$$\llbracket\tau_N({\mathcal E}^N_{\ell,0})\rrbracket=\frac{N-\ell}{2}+\frac{3}{2}-2^{-N+\ell+1}.$$

This gives an exact value for the Carleson constant in the range $0\leq\ell\leq N-2$, complementing the preceding lower and upper estimates and describing the behavior of the postorder rearrangement on these dyadic collections.