Column insertion

Let $T$ be a standard tableau with distinct real entries (Tableaux and standard tableaux) and let $x\mathrm{\notin }T$ be a real number. The column insertion $x\to T$ is defined by the same rules as row insertion with rows replaced by columns: put $x_1:=x$ and consider column $1$. At column $i$: if column $i$ is empty or $x_i$ is larger than every entry of column $i$, append $x_i$ in a new box at the bottom of column $i$ and stop; otherwise let $y_i$ be the topmost entry of column $i$ that is larger than $x_i$, replace that entry by $x_i$, put $x_{i+1}:=y_i$, and continue with column $i+1$. Equivalently, $x\to T=(T^{\mathrm{t}}\leftarrow x{)}^{\mathrm{t}},$ where $T^{\mathrm{t}}$ is the transposed tableau (a standard tableau of shape $[\mathrm{shape}(T{)}^{\mathrm{\prime }}]$, Partitions, English diagrams, and conjugation) and $\leftarrow$ is the row insertion of Row insertion and the bumping route; the equivalence is the observation that transposing a tableau interchanges rows and columns, so "leftmost entry of a row greater than the carried letter" becomes "topmost entry of a column greater than the carried letter".

By the equivalence, the column procedure terminates and $x\to T$ is a standard tableau with entries those of $T$ together with $x$, of shape $[\mathrm{shape}(T)]$ with one box added at the bottom of a column: this is Monotonicity of the bumping route and standardness of the output applied to $T^{\mathrm{t}}$, whose new box transposes back to a box at the bottom of a column of $T$. The route positions weakly decrease from column to column, again by transposing the position bound for row insertion. No choice is used; the procedure is deterministic.