Identifier
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[1]=>[[1]]=>[1]=>[1]=> [2]=>[[1,2]]=>[1,2]=>[2,1]=>0 [1,1]=>[[1],[2]]=>[2,1]=>[1,2]=>1 [3]=>[[1,2,3]]=>[1,2,3]=>[3,2,1]=>00 [2,1]=>[[1,2],[3]]=>[3,1,2]=>[2,1,3]=>01 [1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[1,2,3]=>11 [4]=>[[1,2,3,4]]=>[1,2,3,4]=>[4,3,2,1]=>000 [3,1]=>[[1,2,3],[4]]=>[4,1,2,3]=>[3,2,1,4]=>001 [2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,1,4,3]=>010 [2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[2,1,3,4]=>011 [1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[1,2,3,4]=>111 [5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>[5,4,3,2,1]=>0000 [4,1]=>[[1,2,3,4],[5]]=>[5,1,2,3,4]=>[4,3,2,1,5]=>0001 [3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[3,2,1,5,4]=>0010 [3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[3,2,1,4,5]=>0011 [2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[2,1,4,3,5]=>0101 [2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[2,1,3,4,5]=>0111 [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[1,2,3,4,5]=>1111 [6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[6,5,4,3,2,1]=>00000 [5,1]=>[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[5,4,3,2,1,6]=>00001 [4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[4,3,2,1,6,5]=>00010 [4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[4,3,2,1,5,6]=>00011 [3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[3,2,1,6,5,4]=>00100 [3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[3,2,1,5,4,6]=>00101 [3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[3,2,1,4,5,6]=>00111 [2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[2,1,4,3,6,5]=>01010 [2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[2,1,4,3,5,6]=>01011 [2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[2,1,3,4,5,6]=>01111 [1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[1,2,3,4,5,6]=>11111 [7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[7,6,5,4,3,2,1]=>000000 [6,1]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[6,5,4,3,2,1,7]=>000001 [5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[5,4,3,2,1,7,6]=>000010 [5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[5,4,3,2,1,6,7]=>000011 [4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[4,3,2,1,7,6,5]=>000100 [4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[4,3,2,1,6,5,7]=>000101 [4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[4,3,2,1,5,6,7]=>000111 [3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[3,2,1,6,5,4,7]=>001001 [3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[3,2,1,5,4,7,6]=>001010 [3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[3,2,1,5,4,6,7]=>001011 [3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[3,2,1,4,5,6,7]=>001111 [2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[2,1,4,3,6,5,7]=>010101 [2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[2,1,4,3,5,6,7]=>010111 [2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[2,1,3,4,5,6,7]=>011111 [1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[1,2,3,4,5,6,7]=>111111 [8]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[8,7,6,5,4,3,2,1]=>0000000 [7,1]=>[[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>[7,6,5,4,3,2,1,8]=>0000001 [6,2]=>[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[6,5,4,3,2,1,8,7]=>0000010 [5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[5,4,3,2,1,8,7,6]=>0000100 [5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[5,4,3,2,1,7,6,8]=>0000101 [5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[5,4,3,2,1,6,7,8]=>0000111 [4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[4,3,2,1,8,7,6,5]=>0001000 [4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[4,3,2,1,7,6,5,8]=>0001001 [4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[4,3,2,1,6,5,8,7]=>0001010 [4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[4,3,2,1,6,5,7,8]=>0001011 [4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[4,3,2,1,5,6,7,8]=>0001111 [3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[3,2,1,5,4,7,6,8]=>0010101 [3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[3,2,1,4,5,6,7,8]=>0011111 [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[2,1,4,3,6,5,8,7]=>0101010 [2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[2,1,4,3,6,5,7,8]=>0101011 [2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[2,1,4,3,5,6,7,8]=>0101111 [2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[2,1,3,4,5,6,7,8]=>0111111 [1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[1,2,3,4,5,6,7,8]=>1111111 [5,1,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,1,2,3,4,5]=>[5,4,3,2,1,6,7,8,9,10]=>000011111
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by 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
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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.