OEIS A000296: Partitions without singletons

We explore A000296, the number of ways to partition an n-element set into blocks of size at least two. From the initial terms 1, 0, 1, 1, 4, 11 to diverse combinatorial interpretations—such as complete rhyming schemes, stable partitions of an n-cycle, and permutation patterns where left-to-right maxima coincide with descents—this sequence connects many different viewpoints. We also discuss the exponential generating function, exp(exp(x) - x - 1), and the Bell-number relationship B(n) = A(n) + A(n+1), which together reveal the rich structure underlying partitions without singletons.

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