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Let $w$ be a permutation in $S_n$ which maps to the pair $(P,Q)$ of standard tableaux under the RSK correspondence. Schensted famously proved that the length of the longest increasing subsequence of $w$ equals the length of the first row of the common shape of $P$ and $Q$. Viennot’s geometric reformulation of RSK gives a beautiful pictorial proof of this fact. We describe a quotient ring $R_n$ whose standard monomial theory is governed by Viennot’s construction and longest increasing subsequences. The ring $R_n$ comes from a tool in combinatorial deformation theory called orbit harmonics.