= RSK growth-diagram local rule
{c}
{title2=$\lambda_1=\max(\mu_1,\nu_1)+a,\quad\lambda_r=\max(\mu_r,\nu_r)+\min(\mu_{r-1},\nu_{r-1})-\rho_{r-1}$}
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.
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