TAOCP 7.2.1.3 Exercise 68
Let $\alpha$ be a $t$-combination, so $\alpha$ is a $t$-element subset of ${0,1,\dots,n-1}$.
Section 7.2.1.3: Generating All Combinations
Exercise 68. [M01] [M01] If α is a t-combination, what is (a) ∂tα? (b) ∂t+1α?
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Let $\alpha$ be a $t$-combination, so $\alpha$ is a $t$-element subset of ${0,1,\dots,n-1}$.
The operator $\partial_t \alpha$ produces all $(t-1)$-combinations obtained by deleting one element of $\alpha$. If $\alpha={c_t,\dots,c_1}$, then
$$ \partial_t \alpha={\alpha\setminus{c_j}\mid 1\le j\le t}. $$
Each element of $\partial_t \alpha$ is therefore a $(t-1)$-combination of ${0,1,\dots,n-1}$.
The operator $\partial_{t+1} \alpha$ produces all $(t+1)$-combinations that contain $\alpha$, obtained by adjoining one new element not already in $\alpha$. If $\overline{\alpha}={0,1,\dots,n-1}\setminus \alpha$, then
$$ \partial_{t+1} \alpha={\alpha\cup{x}\mid x\in \overline{\alpha}}. $$
Each element of $\partial_{t+1} \alpha$ is therefore a $(t+1)$-combination containing $\alpha$.