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Identifier

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000736: Semistandard tableaux ⟶ ℤ

Values

[[1,2]] => [[1,2]] => [[1,2]] => [[1,2]] => 2

[[2,2]] => [[2,2]] => [[2,2]] => [[2,2]] => 2

[[1],[2]] => [[1,2]] => [[1,2]] => [[1,2]] => 2

[[1,1,2]] => [[1,1,2]] => [[1,1,2]] => [[1,1,2]] => 2

[[1,2,2]] => [[1,2,2]] => [[1,2,2]] => [[1,2,2]] => 2

[[2,2,2]] => [[2,2,2]] => [[2,2,2]] => [[2,2,2]] => 2

[[1,1],[2]] => [[1,1,2]] => [[1,1,2]] => [[1,1,2]] => 2

[[1,2],[2]] => [[1,2,2]] => [[1,2,2]] => [[1,2,2]] => 2

[[1,1,3]] => [[1,1,3]] => [[1,1,3]] => [[1,1,3]] => 3

[[1,2,3]] => [[1,2,3]] => [[1,2,3]] => [[1,2,3]] => 3

[[1,3,3]] => [[1,3,3]] => [[1,3,3]] => [[1,3,3]] => 3

[[2,2,3]] => [[2,2,3]] => [[2,2,3]] => [[2,2,3]] => 3

[[2,3,3]] => [[2,3,3]] => [[2,3,3]] => [[2,3,3]] => 3

[[3,3,3]] => [[3,3,3]] => [[3,3,3]] => [[3,3,3]] => 3

[[1,1],[3]] => [[1,1,3]] => [[1,1,3]] => [[1,1,3]] => 3

[[1,2],[3]] => [[1,2,3]] => [[1,2,3]] => [[1,2,3]] => 3

[[1,3],[2]] => [[1,2],[3]] => [[1,2,3]] => [[1,2,3]] => 3

[[1,3],[3]] => [[1,3,3]] => [[1,3,3]] => [[1,3,3]] => 3

[[2,2],[3]] => [[2,2,3]] => [[2,2,3]] => [[2,2,3]] => 3

[[2,3],[3]] => [[2,3,3]] => [[2,3,3]] => [[2,3,3]] => 3

[[1],[2],[3]] => [[1,2],[3]] => [[1,2,3]] => [[1,2,3]] => 3

[[1,1,1,2]] => [[1,1,1,2]] => [[1,1,1,2]] => [[1,1,1,2]] => 2

[[1,1,2,2]] => [[1,1,2,2]] => [[1,1,2,2]] => [[1,1,2,2]] => 2

[[1,2,2,2]] => [[1,2,2,2]] => [[1,2,2,2]] => [[1,2,2,2]] => 2

[[2,2,2,2]] => [[2,2,2,2]] => [[2,2,2,2]] => [[2,2,2,2]] => 2

[[1,1,1],[2]] => [[1,1,1,2]] => [[1,1,1,2]] => [[1,1,1,2]] => 2

[[1,1,2],[2]] => [[1,1,2,2]] => [[1,1,2,2]] => [[1,1,2,2]] => 2

[[1,2,2],[2]] => [[1,2,2,2]] => [[1,2,2,2]] => [[1,2,2,2]] => 2

[[1,1],[2,2]] => [[1,1,2,2]] => [[1,1,2,2]] => [[1,1,2,2]] => 2

[[1,1,1,3]] => [[1,1,1,3]] => [[1,1,1,3]] => [[1,1,1,3]] => 3

[[1,1,2,3]] => [[1,1,2,3]] => [[1,1,2,3]] => [[1,1,2,3]] => 3

[[1,1,3,3]] => [[1,1,3,3]] => [[1,1,3,3]] => [[1,1,3,3]] => 3

[[1,2,2,3]] => [[1,2,2,3]] => [[1,2,2,3]] => [[1,2,2,3]] => 3

[[1,2,3,3]] => [[1,2,3,3]] => [[1,2,3,3]] => [[1,2,3,3]] => 3

[[1,3,3,3]] => [[1,3,3,3]] => [[1,3,3,3]] => [[1,3,3,3]] => 3

[[2,2,2,3]] => [[2,2,2,3]] => [[2,2,2,3]] => [[2,2,2,3]] => 3

[[2,2,3,3]] => [[2,2,3,3]] => [[2,2,3,3]] => [[2,2,3,3]] => 3

[[2,3,3,3]] => [[2,3,3,3]] => [[2,3,3,3]] => [[2,3,3,3]] => 3

[[3,3,3,3]] => [[3,3,3,3]] => [[3,3,3,3]] => [[3,3,3,3]] => 3

[[1,1,1],[3]] => [[1,1,1,3]] => [[1,1,1,3]] => [[1,1,1,3]] => 3

[[1,1,2],[3]] => [[1,1,2,3]] => [[1,1,2,3]] => [[1,1,2,3]] => 3

[[1,1,3],[2]] => [[1,1,2],[3]] => [[1,1,2,3]] => [[1,1,2,3]] => 3

[[1,1,3],[3]] => [[1,1,3,3]] => [[1,1,3,3]] => [[1,1,3,3]] => 3

[[1,2,2],[3]] => [[1,2,2,3]] => [[1,2,2,3]] => [[1,2,2,3]] => 3

[[1,2,3],[2]] => [[1,2,2],[3]] => [[1,2,2,3]] => [[1,2,2,3]] => 3

[[1,2,3],[3]] => [[1,2,3,3]] => [[1,2,3,3]] => [[1,2,3,3]] => 3

[[1,3,3],[2]] => [[1,2,3],[3]] => [[1,2,3,3]] => [[1,2,3,3]] => 3

[[1,3,3],[3]] => [[1,3,3,3]] => [[1,3,3,3]] => [[1,3,3,3]] => 3

[[2,2,2],[3]] => [[2,2,2,3]] => [[2,2,2,3]] => [[2,2,2,3]] => 3

[[2,2,3],[3]] => [[2,2,3,3]] => [[2,2,3,3]] => [[2,2,3,3]] => 3

[[2,3,3],[3]] => [[2,3,3,3]] => [[2,3,3,3]] => [[2,3,3,3]] => 3

[[1,1],[2,3]] => [[1,1,2,3]] => [[1,1,2,3]] => [[1,1,2,3]] => 3

[[1,1],[3,3]] => [[1,1,3,3]] => [[1,1,3,3]] => [[1,1,3,3]] => 3

[[1,2],[2,3]] => [[1,2,2,3]] => [[1,2,2,3]] => [[1,2,2,3]] => 3

[[1,2],[3,3]] => [[1,2,3,3]] => [[1,2,3,3]] => [[1,2,3,3]] => 3

[[2,2],[3,3]] => [[2,2,3,3]] => [[2,2,3,3]] => [[2,2,3,3]] => 3

[[1,1],[2],[3]] => [[1,1,2],[3]] => [[1,1,2,3]] => [[1,1,2,3]] => 3

[[1,2],[2],[3]] => [[1,2,2],[3]] => [[1,2,2,3]] => [[1,2,2,3]] => 3

[[1,3],[2],[3]] => [[1,2,3],[3]] => [[1,2,3,3]] => [[1,2,3,3]] => 3

[[1,1,1,1,2]] => [[1,1,1,1,2]] => [[1,1,1,1,2]] => [[1,1,1,1,2]] => 2

[[1,1,1,2,2]] => [[1,1,1,2,2]] => [[1,1,1,2,2]] => [[1,1,1,2,2]] => 2

[[1,1,2,2,2]] => [[1,1,2,2,2]] => [[1,1,2,2,2]] => [[1,1,2,2,2]] => 2

[[1,2,2,2,2]] => [[1,2,2,2,2]] => [[1,2,2,2,2]] => [[1,2,2,2,2]] => 2

[[2,2,2,2,2]] => [[2,2,2,2,2]] => [[2,2,2,2,2]] => [[2,2,2,2,2]] => 2

[[1,1,1,1],[2]] => [[1,1,1,1,2]] => [[1,1,1,1,2]] => [[1,1,1,1,2]] => 2

[[1,1,1,2],[2]] => [[1,1,1,2,2]] => [[1,1,1,2,2]] => [[1,1,1,2,2]] => 2

[[1,1,2,2],[2]] => [[1,1,2,2,2]] => [[1,1,2,2,2]] => [[1,1,2,2,2]] => 2

[[1,2,2,2],[2]] => [[1,2,2,2,2]] => [[1,2,2,2,2]] => [[1,2,2,2,2]] => 2

[[1,1,1],[2,2]] => [[1,1,1,2,2]] => [[1,1,1,2,2]] => [[1,1,1,2,2]] => 2

[[1,1,2],[2,2]] => [[1,1,2,2,2]] => [[1,1,2,2,2]] => [[1,1,2,2,2]] => 2

[[1,1,1,4]] => [[1,1,1,4]] => [[1,1,1,4]] => [[1,1,1,4]] => 4

[[1,1,2,4]] => [[1,1,2,4]] => [[1,1,2,4]] => [[1,1,2,4]] => 4

[[1,1,3,4]] => [[1,1,3,4]] => [[1,1,3,4]] => [[1,1,3,4]] => 4

[[1,1,4,4]] => [[1,1,4,4]] => [[1,1,4,4]] => [[1,1,4,4]] => 4

[[1,2,2,4]] => [[1,2,2,4]] => [[1,2,2,4]] => [[1,2,2,4]] => 4

[[1,2,3,4]] => [[1,2,3,4]] => [[1,2,3,4]] => [[1,2,3,4]] => 4

[[1,2,4,4]] => [[1,2,4,4]] => [[1,2,4,4]] => [[1,2,4,4]] => 4

[[1,3,3,4]] => [[1,3,3,4]] => [[1,3,3,4]] => [[1,3,3,4]] => 4

[[1,3,4,4]] => [[1,3,4,4]] => [[1,3,4,4]] => [[1,3,4,4]] => 4

[[1,4,4,4]] => [[1,4,4,4]] => [[1,4,4,4]] => [[1,4,4,4]] => 4

[[2,2,2,4]] => [[2,2,2,4]] => [[2,2,2,4]] => [[2,2,2,4]] => 4

[[2,2,3,4]] => [[2,2,3,4]] => [[2,2,3,4]] => [[2,2,3,4]] => 4

[[2,2,4,4]] => [[2,2,4,4]] => [[2,2,4,4]] => [[2,2,4,4]] => 4

[[2,3,3,4]] => [[2,3,3,4]] => [[2,3,3,4]] => [[2,3,3,4]] => 4

[[2,3,4,4]] => [[2,3,4,4]] => [[2,3,4,4]] => [[2,3,4,4]] => 4

[[2,4,4,4]] => [[2,4,4,4]] => [[2,4,4,4]] => [[2,4,4,4]] => 4

[[3,3,3,4]] => [[3,3,3,4]] => [[3,3,3,4]] => [[3,3,3,4]] => 4

[[3,3,4,4]] => [[3,3,4,4]] => [[3,3,4,4]] => [[3,3,4,4]] => 4

[[3,4,4,4]] => [[3,4,4,4]] => [[3,4,4,4]] => [[3,4,4,4]] => 4

[[4,4,4,4]] => [[4,4,4,4]] => [[4,4,4,4]] => [[4,4,4,4]] => 4

[[1,1,1],[4]] => [[1,1,1,4]] => [[1,1,1,4]] => [[1,1,1,4]] => 4

[[1,1,2],[4]] => [[1,1,2,4]] => [[1,1,2,4]] => [[1,1,2,4]] => 4

[[1,1,4],[2]] => [[1,1,2],[4]] => [[1,1,2,4]] => [[1,1,2,4]] => 4

[[1,1,3],[4]] => [[1,1,3,4]] => [[1,1,3,4]] => [[1,1,3,4]] => 4

[[1,1,4],[3]] => [[1,1,3],[4]] => [[1,1,3,4]] => [[1,1,3,4]] => 4

[[1,1,4],[4]] => [[1,1,4,4]] => [[1,1,4,4]] => [[1,1,4,4]] => 4

[[1,2,2],[4]] => [[1,2,2,4]] => [[1,2,2,4]] => [[1,2,2,4]] => 4

[[1,2,4],[2]] => [[1,2,2],[4]] => [[1,2,2,4]] => [[1,2,2,4]] => 4

[[1,2,3],[4]] => [[1,2,3,4]] => [[1,2,3,4]] => [[1,2,3,4]] => 4

[[1,2,4],[3]] => [[1,2,3],[4]] => [[1,2,3,4]] => [[1,2,3,4]] => 4

[[1,3,4],[2]] => [[1,2,4],[3]] => [[1,2,3],[4]] => [[1,2,3,4]] => 4

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Description

The last entry in the first row of a semistandard tableau.

Map

catabolism

Description

Remove the first row of the semistandard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry $k$ into tableau $T$ is as follows:
If $k$ is larger than all entries in the first column, place $k$ at the bottom of the first column and the procedure is finished. Otherwise, place $k$ in the first column, replacing the smallest entry, $y$, greater or equal to than $k$. Now insert $y$ into the second column using the same procedure: if $y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place $y$ in the second column, replacing, or 'bumping', the smallest entry, $z$, larger than or equal to $y$. Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).