- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions

Images

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

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

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

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

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

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

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

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

[[1,1,2]] => [1,2,3] => 11 => [1,1,1]

[[1,2,2]] => [1,2,3] => 11 => [1,1,1]

[[2,2,2]] => [1,2,3] => 11 => [1,1,1]

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

[[1,2],[2]] => [2,1,3] => 01 => [2,1]

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

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

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

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

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

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

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

[[1,1,3]] => [1,2,3] => 11 => [1,1,1]

[[1,2,3]] => [1,2,3] => 11 => [1,1,1]

[[1,3,3]] => [1,2,3] => 11 => [1,1,1]

[[2,2,3]] => [1,2,3] => 11 => [1,1,1]

[[2,3,3]] => [1,2,3] => 11 => [1,1,1]

[[3,3,3]] => [1,2,3] => 11 => [1,1,1]

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

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

[[1,3],[2]] => [2,1,3] => 01 => [2,1]

[[1,3],[3]] => [2,1,3] => 01 => [2,1]

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

[[2,3],[3]] => [2,1,3] => 01 => [2,1]

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

[[1,1,1,2]] => [1,2,3,4] => 111 => [1,1,1,1]

[[1,1,2,2]] => [1,2,3,4] => 111 => [1,1,1,1]

[[1,2,2,2]] => [1,2,3,4] => 111 => [1,1,1,1]

[[2,2,2,2]] => [1,2,3,4] => 111 => [1,1,1,1]

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

[[1,1,2],[2]] => [3,1,2,4] => 001 => [3,1]

[[1,2,2],[2]] => [2,1,3,4] => 011 => [2,1,1]

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

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

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

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

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

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

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

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

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

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

[[1,1,4]] => [1,2,3] => 11 => [1,1,1]

[[1,2,4]] => [1,2,3] => 11 => [1,1,1]

[[1,3,4]] => [1,2,3] => 11 => [1,1,1]

[[1,4,4]] => [1,2,3] => 11 => [1,1,1]

[[2,2,4]] => [1,2,3] => 11 => [1,1,1]

[[2,3,4]] => [1,2,3] => 11 => [1,1,1]

[[2,4,4]] => [1,2,3] => 11 => [1,1,1]

[[3,3,4]] => [1,2,3] => 11 => [1,1,1]

[[3,4,4]] => [1,2,3] => 11 => [1,1,1]

[[4,4,4]] => [1,2,3] => 11 => [1,1,1]

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

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

[[1,4],[2]] => [2,1,3] => 01 => [2,1]

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

[[1,4],[3]] => [2,1,3] => 01 => [2,1]

[[1,4],[4]] => [2,1,3] => 01 => [2,1]

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

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

[[2,4],[3]] => [2,1,3] => 01 => [2,1]

[[2,4],[4]] => [2,1,3] => 01 => [2,1]

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

[[3,4],[4]] => [2,1,3] => 01 => [2,1]

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

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

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

[[1,1,1,3]] => [1,2,3,4] => 111 => [1,1,1,1]

[[1,1,2,3]] => [1,2,3,4] => 111 => [1,1,1,1]

[[1,1,3,3]] => [1,2,3,4] => 111 => [1,1,1,1]

[[1,2,2,3]] => [1,2,3,4] => 111 => [1,1,1,1]

[[1,2,3,3]] => [1,2,3,4] => 111 => [1,1,1,1]

[[1,3,3,3]] => [1,2,3,4] => 111 => [1,1,1,1]

[[2,2,2,3]] => [1,2,3,4] => 111 => [1,1,1,1]

[[2,2,3,3]] => [1,2,3,4] => 111 => [1,1,1,1]

[[2,3,3,3]] => [1,2,3,4] => 111 => [1,1,1,1]

[[3,3,3,3]] => [1,2,3,4] => 111 => [1,1,1,1]

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

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

[[1,1,3],[2]] => [3,1,2,4] => 001 => [3,1]

[[1,1,3],[3]] => [3,1,2,4] => 001 => [3,1]

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

[[1,2,3],[2]] => [2,1,3,4] => 011 => [2,1,1]

[[1,2,3],[3]] => [3,1,2,4] => 001 => [3,1]

[[1,3,3],[2]] => [2,1,3,4] => 011 => [2,1,1]

[[1,3,3],[3]] => [2,1,3,4] => 011 => [2,1,1]

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

[[2,2,3],[3]] => [3,1,2,4] => 001 => [3,1]

[[2,3,3],[3]] => [2,1,3,4] => 011 => [2,1,1]

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

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

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

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

>>> Load all 2504 entries. <<<

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

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.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.