First Boundary Model $S^{(1)}$

Definition

The first boundary, or outer frame, of MTMN uses rows and columns $1$ and $N-1$: $$ B^{(1)}_{N,a} := A_{N,a} \cap \bigl((\{1,N-1\}\times\{1,\ldots,N-1\}) \cup (\{1,\ldots,N-1\}\times\{1,N-1\})\bigr). $$ Set $$ S^{(1)}(N,a) := \operatorname{Area}(\operatorname{conv}(B^{(1)}_{N,a})), \qquad S^{(1)}(N) := \sum_{a=0}^{N-1} S^{(1)}(N,a). $$ By construction, $$ B^{(1)}_{N,a} \subseteq A_{N,a}, \qquad S^{(1)}(N,a) \leq S(N,a), \qquad S^{(1)}(N) \leq S(N). $$ So this is an exact lower-bound model rather than a heuristic approximation.

Exact per-residue formula

Theorem (Exact first-boundary formula). For $N \geq 2$, $$ S^{(1)}(N,0)=0, $$ and for $1 \le a \le N-1$, $$ S^{(1)}(N,a)=2(a-1)(N-a-1). $$

The corresponding hull is the parallelogram with vertices $$ (1,a),\quad (a,1),\quad (N-1,N-a),\quad (N-a,N-1). $$ When $a=1$ or $a=N-1$, that parallelogram degenerates and the area is $0$.

Exact total formula

Theorem (Exact total first-boundary sum). For every $N \geq 2$, $$ S^{(1)}(N)=\frac{(N-3)(N-2)(N-1)}{3}. $$

Exact reciprocal constants

Theorem (Exact reciprocal series). $$ \sum_{N=4}^{\infty} \frac{1}{S^{(1)}(N)} = \frac{3}{4}. $$
Theorem (Exact weighted first-boundary series). $$ \sum_{N=4}^{\infty} \frac{N}{S^{(1)}(N)} = \frac{15}{4}. $$

These exact constants show that the first boundary is not only a local geometric model. It also controls the reciprocal and weighted comparison series for the full-MTMN totals.

Figures

First-boundary parallelogram for $N = 11$, $a = 3$.

For $N = 11$ and $a = 3$, the first-boundary parallelogram compared with the full convex hull.

Tags

#definition #theorem #formula