Identifier
Mp00058: to permutationPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00073: Permutations major-index to inversion-number 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]=>[3,2,4,1]=>000 [(1,3),(2,4)]=>[3,4,1,2]=>[3,1,4,2]=>[3,4,1,2]=>000 [(1,4),(2,3)]=>[4,3,2,1]=>[3,2,4,1]=>[4,2,1,3]=>000 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[4,3,5,2,6,1]=>00000 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,1,4,2,6,5]=>[4,5,2,3,6,1]=>00000 [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>[5,3,2,4,6,1]=>00000 [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>[5,3,2,6,1,4]=>00000 [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>[6,3,2,4,1,5]=>00000 [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,2,5,3,6,1]=>[6,3,4,1,2,5]=>00000 [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[4,2,5,1,6,3]=>[5,3,6,1,2,4]=>00000 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[4,5,6,1,2,3]=>00000 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]=>[4,5,2,6,1,3]=>00000 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,5,3,6,4]=>[4,3,5,6,1,2]=>00000 [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[4,3,6,2,1,5]=>00000 [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]=>[4,6,2,3,1,5]=>00000 [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,1,5,3,6,2]=>[4,6,3,1,2,5]=>00000 [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[4,3,5,1,6,2]=>[5,6,2,1,3,4]=>00000 [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[6,4,2,1,3,5]=>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]=>[5,4,6,3,7,2,8,1]=>0000000 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,1,4,2,6,5,8,7]=>[5,6,3,4,7,2,8,1]=>0000000 [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[3,2,5,1,6,4,8,7]=>[6,4,3,7,2,5,8,1]=>0000000 [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[4,2,5,3,7,1,8,6]=>[7,4,5,2,3,8,1,6]=>0000000 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,1,5,2,6,3,8,7]=>[5,6,7,2,3,4,8,1]=>0000000 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,1,5,2,6,4,8,7]=>[5,6,3,7,2,4,8,1]=>0000000 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,5,3,6,4,8,7]=>[5,4,6,7,2,3,8,1]=>0000000 [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[4,1,5,3,7,2,8,6]=>[5,7,4,2,3,8,1,6]=>0000000 [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,1,5,4,7,2,8,6]=>[5,7,3,4,2,8,1,6]=>0000000 [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[3,1,5,4,7,6,8,2]=>[5,8,3,4,2,6,1,7]=>0000000 [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[4,3,6,5,7,1,8,2]=>[7,8,3,2,4,1,5,6]=>0000000 [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1]=>[8,6,3,2,4,1,5,7]=>0000000 [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[5,3,6,4,7,1,8,2]=>[7,8,3,4,1,2,5,6]=>0000000 [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[4,1,6,3,7,5,8,2]=>[5,8,4,2,6,1,3,7]=>0000000 [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,6,4,7,3,8,5]=>[5,4,7,3,8,1,2,6]=>0000000 [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[5,1,6,3,7,2,8,4]=>[5,7,4,8,1,2,3,6]=>0000000 [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[5,3,6,2,7,4,8,1]=>[8,5,3,6,1,2,4,7]=>0000000 [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[5,2,6,3,7,4,8,1]=>[8,4,5,6,1,2,3,7]=>0000000 [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[5,2,6,3,7,1,8,4]=>[7,4,5,8,1,2,3,6]=>0000000 [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[5,2,6,1,7,3,8,4]=>[6,4,7,8,1,2,3,5]=>0000000 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>[5,6,7,8,1,2,3,4]=>0000000 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,1,6,2,7,3,8,5]=>[5,6,7,2,8,1,3,4]=>0000000 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,1,6,2,7,4,8,5]=>[5,6,3,7,8,1,2,4]=>0000000 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,6,3,7,4,8,5]=>[5,4,6,7,8,1,2,3]=>0000000 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,5,3,7,4,8,6]=>[5,4,6,7,2,8,1,3]=>0000000 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,1,5,2,7,4,8,6]=>[5,6,3,7,2,8,1,4]=>0000000 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,1,5,2,7,3,8,6]=>[5,6,7,2,3,8,1,4]=>0000000 [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[4,2,6,1,7,3,8,5]=>[6,4,7,2,8,1,3,5]=>0000000 [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[3,2,6,4,7,5,8,1]=>[8,4,3,5,6,1,2,7]=>0000000 [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[3,2,6,1,7,4,8,5]=>[6,4,3,7,8,1,2,5]=>0000000 [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[3,2,5,1,7,4,8,6]=>[6,4,3,7,2,8,1,5]=>0000000 [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[3,2,4,1,7,5,8,6]=>[6,4,3,5,7,8,1,2]=>0000000 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,1,4,2,7,5,8,6]=>[5,6,3,4,7,8,1,2]=>0000000 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,5,8,6]=>[5,4,6,3,7,8,1,2]=>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]=>[5,4,6,3,8,2,1,7]=>0000000 [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[3,1,4,2,7,6,8,5]=>[5,6,3,4,8,2,1,7]=>0000000 [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>[6,4,3,5,8,2,1,7]=>0000000 [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,2,5,1,7,6,8,4]=>[6,4,3,8,2,5,1,7]=>0000000 [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[4,2,6,1,7,5,8,3]=>[6,4,8,2,5,1,3,7]=>0000000 [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[3,1,5,2,7,6,8,4]=>[5,6,3,8,2,4,1,7]=>0000000 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[4,1,6,2,7,5,8,3]=>[5,6,8,2,4,1,3,7]=>0000000 [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[5,1,6,4,7,2,8,3]=>[5,7,8,3,1,2,4,6]=>0000000 [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[5,8,6,2,3,1,4,7]=>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]=>[8,6,4,2,1,3,5,7]=>0000000
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
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.