Doubly ordered set partitions

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Exponential generating functions

Set partitions	Any # of blocks	Odd # of blocks	Even # of blocks
Any block sizes	$e^{e^x - 1}$	$\sinh(e^x - 1)$	$\cosh(e^x - 1)$
Odd block sizes	$e^{\sinh x}$	$\sinh(\sinh x)$	$\cosh(\sinh x)$
Even block sizes	$e^{\cosh x - 1}$	$\sinh(\cosh x - 1)$	$\cosh(\cosh x - 1)$

References

Flajolet, Sedgewick: Analytic Combinatorics

Comtet: Advanced Combinatorics

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