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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces A set-valued tableaux $T$ where $\phi (p(T)) \not =r(\phi (T))$, as rotation of $\phi (T)$ changes the matching's type from $\mathaccentV {vec}17E{b} = (3,2,2)$ to $\mathaccentV {vec}17E{b}'=(2,2,3)$.\relax }}{9}{figure.caption.6}}
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\newlabel{tab: jdq enumeration}{{1}{12}{$| \widetilde {\svt } (n^2,\rho | = | \widetilde {\mathcal {M}}_n^k |$ for $\rho $ the row-constant density $\rho _{1,j} = k-1$, $\rho _{2,j} = 1$.\relax }{table.caption.9}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Non-crossing $3$-point matchings $M_1,M_2 \in \mathcal  {M}_7^3$ of order $7$, constructed from their sub-matchings $q_3(M_1),q_3(M_2)$. Notice that $r(M_1) = M_2$, implying that their associated set-valued tableaux are jeu de taquin equivalent.\relax }}{13}{figure.caption.10}}
\newlabel{fig: sub-matching example}{{9}{13}{Non-crossing $3$-point matchings $M_1,M_2 \in \mathcal {M}_7^3$ of order $7$, constructed from their sub-matchings $q_3(M_1),q_3(M_2)$. Notice that $r(M_1) = M_2$, implying that their associated set-valued tableaux are jeu de taquin equivalent.\relax }{figure.caption.10}{}}
\newlabel{thm: sub-matching enumeration}{{7}{14}{}{theorem.7}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {4}Acknowledgments}{17}{section.4}}
\@writefile{toc}{\contentsline {section}{\numberline {A}On rational non-crossing matchings}{17}{appendix.A}}
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\newlabel{fig: rational Catalan matching comparison}{{10}{18}{A standard set-valued Young tableaux of density $\vec {b}_{(5,8)}$, the corresponding $(5,8)$-Dyck path (with lasers in red), and a comparison of the matchings that result from our methodology (TOP) and the methods of Armstrong, Rhoades, and Williams (BOTTOM).\relax }{figure.caption.11}{}}
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