  We investigate and characterize the dynamics of operator growth in irrational
two-dimensional conformal field theories. By employing the oscillator
realization of the Virasoro algebra and CFT states, we systematically implement
the Lanczos algorithm and evaluate the Krylov complexity of simple operators
(primaries and the stress tensor) under a unitary evolution protocol. Evolution
of primary operators proceeds as a flow into the 'bath of descendants' of the
Verma module. These descendants are labeled by integer partitions and have a
one-to-one map to Young diagrams. This relationship allows us to rigorously
formulate operator growth as paths spreading along the Young's lattice. We
extract quantitative features of these paths and also identify the one that
saturates the conjectured upper bound on operator growth.
