Identifier
Mp00058: to permutationPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00067: Permutations Foata bijectionPermutations
Mp00114: Permutations connectivity setBinary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1]=>0 [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[4,2,1,3]=>000 [(1,3),(2,4)]=>[3,4,1,2]=>[4,1,3,2]=>[4,1,3,2]=>000 [(1,4),(2,3)]=>[4,3,2,1]=>[3,2,4,1]=>[3,2,4,1]=>000 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[6,4,2,1,3,5]=>00000 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,1,3,2,6,5]=>[6,4,1,3,2,5]=>00000 [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>[6,3,2,4,1,5]=>00000 [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,2,6,1,5,4]=>[6,3,2,1,5,4]=>00000 [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>[5,3,2,4,6,1]=>00000 [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[5,2,4,3,6,1]=>[5,2,4,3,6,1]=>00000 [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[6,2,4,1,5,3]=>[4,2,6,5,1,3]=>00000 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,5,2,4,3]=>[6,1,5,2,4,3]=>00000 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[6,1,3,2,5,4]=>[6,1,5,3,2,4]=>00000 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>[6,2,5,1,3,4]=>00000 [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[5,4,2,1,6,3]=>00000 [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[5,4,6,1,3,2]=>[5,1,4,6,3,2]=>00000 [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[5,1,6,3,4,2]=>[5,1,6,3,4,2]=>00000 [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[4,3,6,1,5,2]=>[4,3,6,1,5,2]=>00000 [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[4,3,5,2,6,1]=>00000 [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[8,6,4,2,1,3,5,7]=>0000000 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,1,3,2,6,5,8,7]=>[8,6,4,1,3,2,5,7]=>0000000 [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[3,2,4,1,6,5,8,7]=>[8,6,3,2,4,1,5,7]=>0000000 [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[3,2,5,4,8,1,7,6]=>[8,5,3,2,4,1,7,6]=>0000000 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,1,5,2,4,3,8,7]=>[8,6,1,5,2,4,3,7]=>0000000 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[6,1,3,2,5,4,8,7]=>[8,6,1,5,3,2,4,7]=>0000000 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>[8,6,2,5,1,3,4,7]=>0000000 [(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[4,3,8,1,7,2,6,5]=>[8,4,3,1,7,2,6,5]=>0000000 [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[5,1,7,6,8,3,4,2]=>[7,1,5,6,3,8,4,2]=>0000000 [(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[4,3,7,1,8,5,6,2]=>[4,7,3,8,1,5,6,2]=>0000000 [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[4,3,6,5,8,1,7,2]=>[6,4,3,5,8,1,7,2]=>0000000 [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[7,3,5,1,8,4,6,2]=>[3,7,5,8,1,4,6,2]=>0000000 [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,8,4,6,3,7,5]=>[8,2,6,4,1,7,3,5]=>0000000 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,1,7,2,6,3,5,4]=>[8,1,7,2,6,3,5,4]=>0000000 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[8,1,7,2,4,3,6,5]=>[8,1,7,2,6,4,3,5]=>0000000 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[8,1,3,2,7,4,6,5]=>[8,1,7,3,6,2,4,5]=>0000000 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,7,4,6,5]=>[8,2,7,1,6,3,4,5]=>0000000 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,8,3,5,4,7,6]=>[8,2,7,5,1,3,4,6]=>0000000 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[8,1,3,2,5,4,7,6]=>[8,1,7,5,3,2,4,6]=>0000000 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[8,1,5,2,4,3,7,6]=>[8,1,7,5,2,4,3,6]=>0000000 [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[8,2,4,1,5,3,7,6]=>[8,4,2,5,7,1,3,6]=>0000000 [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,2,4,1,7,3,6,5]=>[8,4,2,1,7,6,3,5]=>0000000 [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[3,2,4,1,8,5,7,6]=>[8,3,7,2,4,1,5,6]=>0000000 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>[8,4,7,1,3,2,5,6]=>0000000 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>[8,4,7,2,1,3,5,6]=>0000000 [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,8,5]=>[7,6,4,2,1,3,8,5]=>0000000 [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[3,2,6,5,8,1,7,4]=>[6,5,3,2,8,1,7,4]=>0000000 [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[7,2,4,1,8,5,6,3]=>[7,2,4,8,5,1,6,3]=>0000000 [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[7,1,8,2,6,4,5,3]=>[7,1,8,2,6,4,5,3]=>0000000 [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[5,4,7,2,6,3,8,1]=>[5,4,7,2,6,3,8,1]=>0000000 [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[5,4,8,2,6,1,7,3]=>[5,4,6,2,8,7,1,3]=>0000000 [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[6,1,8,4,7,2,5,3]=>[6,1,8,4,7,2,5,3]=>0000000 [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]=>[5,4,6,3,7,2,8,1]=>0000000 [(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[2,1,4,3,6,5,8,7,10,9]=>[10,8,6,4,2,1,3,5,7,9]=>000000000 [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[12,10,8,6,4,2,1,3,5,7,9,11]=>00000000000
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
• If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
• If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
• $1$
• $|1|4 \to 14$
• $|14|2 \to 412$
• $|4|1|2|5 \to 4125$
• $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
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.