- FindStat

Identifier

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

[[1,2]] => [[1,2]] => [1,2] => [2,1] => 0

[[2,2]] => [[2,2]] => [1,2] => [2,1] => 0

[[1],[2]] => [[1,2]] => [1,2] => [2,1] => 0

[[1,3]] => [[1,3]] => [1,2] => [2,1] => 0

[[2,3]] => [[2,3]] => [1,2] => [2,1] => 0

[[3,3]] => [[3,3]] => [1,2] => [2,1] => 0

[[1],[3]] => [[1,3]] => [1,2] => [2,1] => 0

[[2],[3]] => [[2,3]] => [1,2] => [2,1] => 0

[[1,1,2]] => [[1,1,2]] => [1,2,3] => [2,3,1] => 00

[[1,2,2]] => [[1,2,2]] => [1,2,3] => [2,3,1] => 00

[[2,2,2]] => [[2,2,2]] => [1,2,3] => [2,3,1] => 00

[[1,1],[2]] => [[1,1,2]] => [1,2,3] => [2,3,1] => 00

[[1,2],[2]] => [[1,2,2]] => [1,2,3] => [2,3,1] => 00

[[1,4]] => [[1,4]] => [1,2] => [2,1] => 0

[[2,4]] => [[2,4]] => [1,2] => [2,1] => 0

[[3,4]] => [[3,4]] => [1,2] => [2,1] => 0

[[4,4]] => [[4,4]] => [1,2] => [2,1] => 0

[[1],[4]] => [[1,4]] => [1,2] => [2,1] => 0

[[2],[4]] => [[2,4]] => [1,2] => [2,1] => 0

[[3],[4]] => [[3,4]] => [1,2] => [2,1] => 0

[[1,1,3]] => [[1,1,3]] => [1,2,3] => [2,3,1] => 00

[[1,2,3]] => [[1,2,3]] => [1,2,3] => [2,3,1] => 00

[[1,3,3]] => [[1,3,3]] => [1,2,3] => [2,3,1] => 00

[[2,2,3]] => [[2,2,3]] => [1,2,3] => [2,3,1] => 00

[[2,3,3]] => [[2,3,3]] => [1,2,3] => [2,3,1] => 00

[[3,3,3]] => [[3,3,3]] => [1,2,3] => [2,3,1] => 00

[[1,1],[3]] => [[1,1,3]] => [1,2,3] => [2,3,1] => 00

[[1,2],[3]] => [[1,2,3]] => [1,2,3] => [2,3,1] => 00

[[1,3],[2]] => [[1,2],[3]] => [3,1,2] => [3,1,2] => 00

[[1,3],[3]] => [[1,3,3]] => [1,2,3] => [2,3,1] => 00

[[2,2],[3]] => [[2,2,3]] => [1,2,3] => [2,3,1] => 00

[[2,3],[3]] => [[2,3,3]] => [1,2,3] => [2,3,1] => 00

[[1],[2],[3]] => [[1,2],[3]] => [3,1,2] => [3,1,2] => 00

[[1,1,1,2]] => [[1,1,1,2]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1,2,2]] => [[1,1,2,2]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,2,2,2]] => [[1,2,2,2]] => [1,2,3,4] => [2,3,4,1] => 000

[[2,2,2,2]] => [[2,2,2,2]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1,1],[2]] => [[1,1,1,2]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1,2],[2]] => [[1,1,2,2]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,2,2],[2]] => [[1,2,2,2]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1],[2,2]] => [[1,1,2,2]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,5]] => [[1,5]] => [1,2] => [2,1] => 0

[[2,5]] => [[2,5]] => [1,2] => [2,1] => 0

[[3,5]] => [[3,5]] => [1,2] => [2,1] => 0

[[4,5]] => [[4,5]] => [1,2] => [2,1] => 0

[[5,5]] => [[5,5]] => [1,2] => [2,1] => 0

[[1],[5]] => [[1,5]] => [1,2] => [2,1] => 0

[[2],[5]] => [[2,5]] => [1,2] => [2,1] => 0

[[3],[5]] => [[3,5]] => [1,2] => [2,1] => 0

[[4],[5]] => [[4,5]] => [1,2] => [2,1] => 0

[[1,1,4]] => [[1,1,4]] => [1,2,3] => [2,3,1] => 00

[[1,2,4]] => [[1,2,4]] => [1,2,3] => [2,3,1] => 00

[[1,3,4]] => [[1,3,4]] => [1,2,3] => [2,3,1] => 00

[[1,4,4]] => [[1,4,4]] => [1,2,3] => [2,3,1] => 00

[[2,2,4]] => [[2,2,4]] => [1,2,3] => [2,3,1] => 00

[[2,3,4]] => [[2,3,4]] => [1,2,3] => [2,3,1] => 00

[[2,4,4]] => [[2,4,4]] => [1,2,3] => [2,3,1] => 00

[[3,3,4]] => [[3,3,4]] => [1,2,3] => [2,3,1] => 00

[[3,4,4]] => [[3,4,4]] => [1,2,3] => [2,3,1] => 00

[[4,4,4]] => [[4,4,4]] => [1,2,3] => [2,3,1] => 00

[[1,1],[4]] => [[1,1,4]] => [1,2,3] => [2,3,1] => 00

[[1,2],[4]] => [[1,2,4]] => [1,2,3] => [2,3,1] => 00

[[1,4],[2]] => [[1,2],[4]] => [3,1,2] => [3,1,2] => 00

[[1,3],[4]] => [[1,3,4]] => [1,2,3] => [2,3,1] => 00

[[1,4],[3]] => [[1,3],[4]] => [3,1,2] => [3,1,2] => 00

[[1,4],[4]] => [[1,4,4]] => [1,2,3] => [2,3,1] => 00

[[2,2],[4]] => [[2,2,4]] => [1,2,3] => [2,3,1] => 00

[[2,3],[4]] => [[2,3,4]] => [1,2,3] => [2,3,1] => 00

[[2,4],[3]] => [[2,3],[4]] => [3,1,2] => [3,1,2] => 00

[[2,4],[4]] => [[2,4,4]] => [1,2,3] => [2,3,1] => 00

[[3,3],[4]] => [[3,3,4]] => [1,2,3] => [2,3,1] => 00

[[3,4],[4]] => [[3,4,4]] => [1,2,3] => [2,3,1] => 00

[[1],[2],[4]] => [[1,2],[4]] => [3,1,2] => [3,1,2] => 00

[[1],[3],[4]] => [[1,3],[4]] => [3,1,2] => [3,1,2] => 00

[[2],[3],[4]] => [[2,3],[4]] => [3,1,2] => [3,1,2] => 00

[[1,1,1,3]] => [[1,1,1,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1,2,3]] => [[1,1,2,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1,3,3]] => [[1,1,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,2,2,3]] => [[1,2,2,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,2,3,3]] => [[1,2,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,3,3,3]] => [[1,3,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[2,2,2,3]] => [[2,2,2,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[2,2,3,3]] => [[2,2,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[2,3,3,3]] => [[2,3,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[3,3,3,3]] => [[3,3,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1,1],[3]] => [[1,1,1,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1,2],[3]] => [[1,1,2,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1,3],[2]] => [[1,1,2],[3]] => [4,1,2,3] => [3,4,1,2] => 000

[[1,1,3],[3]] => [[1,1,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,2,2],[3]] => [[1,2,2,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,2,3],[2]] => [[1,2,2],[3]] => [4,1,2,3] => [3,4,1,2] => 000

[[1,2,3],[3]] => [[1,2,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,3,3],[2]] => [[1,2,3],[3]] => [3,1,2,4] => [3,4,2,1] => 000

[[1,3,3],[3]] => [[1,3,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[2,2,2],[3]] => [[2,2,2,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[2,2,3],[3]] => [[2,2,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[2,3,3],[3]] => [[2,3,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1],[2,3]] => [[1,1,2,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,1],[3,3]] => [[1,1,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,2],[2,3]] => [[1,2,2,3]] => [1,2,3,4] => [2,3,4,1] => 000

[[1,2],[3,3]] => [[1,2,3,3]] => [1,2,3,4] => [2,3,4,1] => 000

>>> Load all 2468 entries. <<<

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).

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).

Map

Kreweras complement

Description

Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.

Map

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.