= Trace and odd columns in symmetric RSK
{title2=$\operatorname{tr}A=\lambda_1-\lambda_2+\lambda_3-\cdots$}
For a symmetric nonnegative integer <matrix>, the two neighboring prefix shapes along the diagonal are equal. In the <RSK growth-diagram local rule>, set $\mu=\nu$ and take alternating sums of row lengths. The two occurrences of the alternating sum of $\mu$ cancel, leaving $o(\lambda)=o(\rho)+a_{ii}$. Induction along the diagonal proves the formula. Each column contributes one to this alternating sum exactly when its length is odd.
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