Combinatorics I have been chasing for years: ($U$, 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.

People proving it using mechanical-memory idea:

• $\textrm{Let }m>k: \sum\limits_{l=0}^{k}\sum\limits_{j=l}^m=\sum\limits_{j=0}^m\sum\limits_{l=0}^{\min(j,k)}$ // Independentness of addition.