Row insertion ($P\leftarrow x$)

Insert $x$ into the first row of a Young tableau by replacing its leftmost entry strictly greater than $x$, and carry the displaced entry into the next row. If no entry is greater, append $x$ and stop. The procedure preserves a semistandard Young tableau; when all entries are distinct it preserves a near Young tableau. It underlies the Robinson–Schensted correspondence.

During row insertion into a near Young tableau, the successive carried entries strictly increase while the columns in which they are bumped weakly decrease. Every old cell keeps its entry or receives a smaller one. These facts follow from increasing rows and strictly increasing columns.