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For groups represented as permutations. Group transitivity, rank 3 groups, orbits and suborbits, stablizers, permutation characters, primitivity are all on-topic.

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A sum over partitions involving "subpartitions"

Consider the generating functions $$ R_k(u) = \sum_{j_1,\ldots,j_k\geq 0} u^{j_1 + 2 j_2 + \cdots + k j_k} \prod_{t=1}^k \frac 1{j_t! t^{j_t}} f_t(j_1,\ldots,j_t). $$ I will prove by induction on $k$ …
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