($U$, the universe, is defined by $S_0$)

• $\textrm{Exactly-IEP}_U(k)=\sum\limits_{j=k}^m(-1)^{j-k}\color{blue}{\binom{j}{k}}S_j.\quad$ // All I have for you is a word, TENET.
• $\textrm{At-Least-IEP}_U(k)=\sum\limits_{j=k}^m(-1)^{j-k}\color{blue}{\binom{j-1}{j-k}}S_j.\quad$ // Pascal Explosion.
• $\textrm{Grid-Scan}(k,m)=\sum\limits_{l=0}^{k}\sum\limits_{\color{blue}j=l}^m=\sum\limits_{j=0}^m\sum\limits_{l=0}^{\color{blue}\min(j,k)}.\quad$ // Two sides, one promise.