Identifier
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ 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]=>[1,3,2]=>10 [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]=>[1,4,3,2]=>100 [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]=>[1,2,4,3]=>110 [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]=>[1,5,4,3,2]=>1000 [3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[2,1,5,4,3]=>0100 [3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[1,2,5,4,3]=>1100 [2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[1,3,2,5,4]=>1010 [2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[1,2,3,5,4]=>1110 [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]=>[1,6,5,4,3,2]=>10000 [4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[2,1,6,5,4,3]=>01000 [4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[1,2,6,5,4,3]=>11000 [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]=>[1,3,2,6,5,4]=>10100 [3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[1,2,3,6,5,4]=>11100 [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]=>[1,2,4,3,6,5]=>11010 [2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[1,2,3,4,6,5]=>11110 [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]=>[1,7,6,5,4,3,2]=>100000 [5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[2,1,7,6,5,4,3]=>010000 [5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[1,2,7,6,5,4,3]=>110000 [4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[3,2,1,7,6,5,4]=>001000 [4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[1,3,2,7,6,5,4]=>101000 [4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[1,2,3,7,6,5,4]=>111000 [3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[1,4,3,2,7,6,5]=>100100 [3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[2,1,4,3,7,6,5]=>010100 [3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[1,2,4,3,7,6,5]=>110100 [3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[1,2,3,4,7,6,5]=>111100 [2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[1,3,2,5,4,7,6]=>101010 [2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[1,2,3,5,4,7,6]=>111010 [2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[1,2,3,4,5,7,6]=>111110 [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]=>[1,8,7,6,5,4,3,2]=>1000000 [6,2]=>[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[2,1,8,7,6,5,4,3]=>0100000 [6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[8,7,1,2,3,4,5,6]=>[1,2,8,7,6,5,4,3]=>1100000 [5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[3,2,1,8,7,6,5,4]=>0010000 [5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[1,3,2,8,7,6,5,4]=>1010000 [5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[1,2,3,8,7,6,5,4]=>1110000 [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]=>[1,4,3,2,8,7,6,5]=>1001000 [4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[2,1,4,3,8,7,6,5]=>0101000 [4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[1,2,4,3,8,7,6,5]=>1101000 [4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[1,2,3,4,8,7,6,5]=>1111000 [3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[1,3,2,5,4,8,7,6]=>1010100 [3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[8,7,6,4,5,1,2,3]=>[1,2,3,5,4,8,7,6]=>1110100 [3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[1,2,3,4,5,8,7,6]=>1111100 [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]=>[1,2,4,3,6,5,8,7]=>1101010 [2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[1,2,3,4,6,5,8,7]=>1111010 [2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[1,2,3,4,5,6,8,7]=>1111110 [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]=>[1,2,3,4,5,10,9,8,7,6]=>111110000
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
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
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.