 RSK growth-diagram local rule

At one matrix entry $a$, let $\rho ,\mu ,\nu ,\lambda$ be the shapes of the northwest, northeast, southwest and southeast matrix prefixes under the RSK correspondence. The displayed row-length recurrence expresses merged insertion paths: the new entry supplies the first-row carry, and the overlap of the two incoming row extensions beyond the northwest shape is bumped into the next row. Zero boundary shapes determine the whole diagram. Transposing the matrix exchanges $\mu ,\nu$, leaving the rule invariant.