Identifier

Mp00033:
Dyck paths

Mp00081: Standard tableaux

Mp00114: Permutations

**—**to two-row standard tableau⟶ Standard tableauxMp00081: Standard tableaux

**—**reading word permutation⟶ PermutationsMp00114: Permutations

**—**connectivity set⟶ Binary words
Images

=>

Cc0005;cc-rep-0Cc0007;cc-rep-1

[1,0]=>[[1],[2]]=>[2,1]=>0
[1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>000
[1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>000
[1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>00000
[1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>00000
[1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>00000
[1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>00000
[1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>00000
[1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>0000000
[1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[2,6,7,8,1,3,4,5]=>0000000
[1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>0000000
[1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[4,5,6,8,1,2,3,7]=>0000000
[1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>0000000
[1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>0000000

Map

**to two-row standard tableau**

Description

Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.

Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.

Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.

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

**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.

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.

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