- FindStat

Identifier

Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000724: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3,4,5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => 5

[[1,3,4,5],[2]] => [[1,2,4,5],[3]] => [3,1,2,4,5] => 5

[[1,2,4,5],[3]] => [[1,2,3,5],[4]] => [4,1,2,3,5] => 5

[[1,2,3,5],[4]] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 4

[[1,2,3,4],[5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => 5

[[1,3,5],[2,4]] => [[1,2,4],[3,5]] => [3,5,1,2,4] => 4

[[1,2,5],[3,4]] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 4

[[1,3,4],[2,5]] => [[1,2,4,5],[3]] => [3,1,2,4,5] => 5

[[1,2,4],[3,5]] => [[1,2,3,5],[4]] => [4,1,2,3,5] => 5

[[1,2,3],[4,5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => 5

[[1,4,5],[2],[3]] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => 5

[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => 4

[[1,2,5],[3],[4]] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 3

[[1,3,4],[2],[5]] => [[1,2,4,5],[3]] => [3,1,2,4,5] => 5

[[1,2,4],[3],[5]] => [[1,2,3,5],[4]] => [4,1,2,3,5] => 5

[[1,2,3],[4],[5]] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 4

[[1,4],[2,5],[3]] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => 5

[[1,3],[2,5],[4]] => [[1,2,4,5],[3]] => [3,1,2,4,5] => 5

[[1,2],[3,5],[4]] => [[1,2,3,5],[4]] => [4,1,2,3,5] => 5

[[1,3],[2,4],[5]] => [[1,2,4],[3,5]] => [3,5,1,2,4] => 4

[[1,2],[3,4],[5]] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 4

[[1,5],[2],[3],[4]] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2

[[1,4],[2],[3],[5]] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => 5

[[1,3],[2],[4],[5]] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => 4

[[1,2],[3],[4],[5]] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 3

[[1],[2],[3],[4],[5]] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2

[[1,2,3,4,5,6]] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 6

[[1,3,4,5,6],[2]] => [[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => 6

[[1,2,4,5,6],[3]] => [[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => 6

[[1,2,3,5,6],[4]] => [[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => 6

[[1,2,3,4,6],[5]] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 5

[[1,2,3,4,5],[6]] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 6

[[1,3,5,6],[2,4]] => [[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => 6

[[1,2,5,6],[3,4]] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 4

[[1,3,4,6],[2,5]] => [[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => 5

[[1,2,4,6],[3,5]] => [[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => 5

[[1,2,3,6],[4,5]] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 5

[[1,3,4,5],[2,6]] => [[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => 6

[[1,2,4,5],[3,6]] => [[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => 6

[[1,2,3,5],[4,6]] => [[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => 6

[[1,2,3,4],[5,6]] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 6

[[1,4,5,6],[2],[3]] => [[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => 6

[[1,3,5,6],[2],[4]] => [[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => 6

[[1,2,5,6],[3],[4]] => [[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => 6

[[1,3,4,6],[2],[5]] => [[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => 5

[[1,2,4,6],[3],[5]] => [[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => 5

[[1,2,3,6],[4],[5]] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 4

[[1,3,4,5],[2],[6]] => [[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => 6

[[1,2,4,5],[3],[6]] => [[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => 6

[[1,2,3,5],[4],[6]] => [[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => 6

[[1,2,3,4],[5],[6]] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 5

[[1,3,5],[2,4,6]] => [[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => 6

[[1,2,5],[3,4,6]] => [[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => 6

[[1,3,4],[2,5,6]] => [[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => 6

[[1,2,4],[3,5,6]] => [[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => 6

[[1,2,3],[4,5,6]] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 6

[[1,4,6],[2,5],[3]] => [[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => 5

[[1,3,6],[2,5],[4]] => [[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => 5

[[1,2,6],[3,5],[4]] => [[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => 5

[[1,3,6],[2,4],[5]] => [[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => 4

[[1,2,6],[3,4],[5]] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 4

[[1,4,5],[2,6],[3]] => [[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => 6

[[1,3,5],[2,6],[4]] => [[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => 6

[[1,2,5],[3,6],[4]] => [[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => 6

[[1,3,4],[2,6],[5]] => [[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => 6

[[1,2,4],[3,6],[5]] => [[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => 6

[[1,2,3],[4,6],[5]] => [[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => 6

[[1,3,5],[2,4],[6]] => [[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => 6

[[1,2,5],[3,4],[6]] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 4

[[1,3,4],[2,5],[6]] => [[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => 5

[[1,2,4],[3,5],[6]] => [[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => 5

[[1,2,3],[4,5],[6]] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 5

[[1,5,6],[2],[3],[4]] => [[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => 6

[[1,4,6],[2],[3],[5]] => [[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => 5

[[1,3,6],[2],[4],[5]] => [[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => 4

[[1,2,6],[3],[4],[5]] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 3

[[1,4,5],[2],[3],[6]] => [[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => 6

[[1,3,5],[2],[4],[6]] => [[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => 6

[[1,2,5],[3],[4],[6]] => [[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => 6

[[1,3,4],[2],[5],[6]] => [[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => 5

[[1,2,4],[3],[5],[6]] => [[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => 5

[[1,2,3],[4],[5],[6]] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 4

[[1,4],[2,5],[3,6]] => [[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => 5

[[1,3],[2,5],[4,6]] => [[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => 5

[[1,2],[3,5],[4,6]] => [[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => 5

>>> Load all 122 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation.
Associate an increasing binary tree to the permutation using Mp00061to increasing tree. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1].
Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the greater neighbor of the maximum (St000060The greater neighbor of the maximum.), see also [3].

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.

Map

catabolism

Description

Remove the first row of the standard 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 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 $y$. Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).