Tabloid and column orders for Specht straightening

Fix a partition $\lambda ⊢n$. If $T$ is a $\lambda$-tabloid, let $r_T(a)$ be the row containing label $a$. For distinct tabloids $T,U$, let $m$ be the largest label for which $r_T(m)\mathrm{\ne }{r}_U(m)$. Define $T>U⟺r_T(m)>r_U(m)\mathrm{.}$ This is the reverse lexicographic order on the row-index vectors $(r_T(n),\dots ,r_T(1))$: any two distinct vectors have a largest differing coordinate, and the row numbers there are comparable. Thus it is a finite strict total order on the $\lambda$-tabloids.

A $\lambda$-tableau is column-standard when its entries strictly increase down each column. For a column-standard tableau $t$, let $c_t(a)$ be the column containing label $a$. Since entries within each column are sorted, the assignment $a\mapsto c_t(a)$ determines $t$ uniquely. For distinct column-standard tableaux $t,u$, let $m$ be the largest label with $c_t(m)\mathrm{\ne }{c}_u(m)$, and define

$$t\prec u⟺c_t(m)<c_u(m)\mathrm{.}$$

This is a finite strict total order on the column-standard tableaux; in words, the largest label assigned to different columns is farther left in $t$ than in $u$. For $\lambda =\mathrm{\varnothing }$, there is one tabloid and one tableau, and each is the sole element of its order.