  In two spacetime dimensions the Virasoro heavy-heavy-light-light (HHLL)
vacuum block in a certain limit is governed by the Catalan numbers. The
equation for their generating function can be generalized to a differential
equation which the logarithm of the block satisfies. We show that a similar
story holds for the HHLL ${\cal W}_N$ vacuum blocks, where a suitable
generalization of the Catalan numbers plays the main role. Moreover, the ${\cal
W}_N$ blocks have the same form as the stress tensor sector of HHLL near
lightcone conformal correlators in $2(N-1)$ spacetime dimensions. In the latter
case the Catalan numbers are generalized to the numbers of linear extensions of
certain partially ordered sets.
