Identifier
Mp00058: to permutationPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1] [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3] [(1,3),(2,4)]=>[3,4,1,2]=>[3,1,4,2]=>[4,3,1,2] [(1,4),(2,3)]=>[4,3,2,1]=>[3,2,4,1]=>[4,3,2,1] [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5] [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,1,4,2,6,5]=>[4,3,1,2,6,5] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>[4,3,2,1,6,5] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>[6,3,2,5,1,4] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>[6,3,2,5,4,1] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,2,5,3,6,1]=>[6,5,4,2,3,1] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[4,2,5,1,6,3]=>[6,5,4,2,1,3] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[6,5,4,1,2,3] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]=>[6,3,1,5,2,4] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,5,3,6,4]=>[2,1,6,5,3,4] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[2,1,6,5,4,3] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]=>[6,3,1,5,4,2] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,1,5,3,6,2]=>[6,5,4,1,3,2] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[4,3,5,1,6,2]=>[6,5,4,3,1,2] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[6,5,4,3,2,1] [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7] [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,1,4,2,6,5,8,7]=>[4,3,1,2,6,5,8,7] [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[3,2,4,1,6,5,8,7]=>[4,3,2,1,6,5,8,7] [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[3,2,5,1,6,4,8,7]=>[6,3,2,5,1,4,8,7] [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[3,2,5,4,6,1,8,7]=>[6,3,2,5,4,1,8,7] [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[3,2,5,4,7,1,8,6]=>[8,3,2,5,4,7,1,6] [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[3,2,5,4,7,6,8,1]=>[8,3,2,5,4,7,6,1] [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[4,2,5,3,7,6,8,1]=>[8,5,4,2,3,7,6,1] [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[4,2,5,3,7,1,8,6]=>[8,5,4,2,3,7,1,6] [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[4,2,5,3,6,1,8,7]=>[6,5,4,2,3,1,8,7] [(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[4,2,5,1,6,3,8,7]=>[6,5,4,2,1,3,8,7] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,1,5,2,6,3,8,7]=>[6,5,4,1,2,3,8,7] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,1,5,2,6,4,8,7]=>[6,3,1,5,2,4,8,7] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,5,3,6,4,8,7]=>[2,1,6,5,3,4,8,7] [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,5,4,6,3,8,7]=>[2,1,6,5,4,3,8,7] [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[3,1,5,4,6,2,8,7]=>[6,3,1,5,4,2,8,7] [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[4,1,5,3,6,2,8,7]=>[6,5,4,1,3,2,8,7] [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[4,3,5,2,6,1,8,7]=>[6,5,4,3,2,1,8,7] [(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[4,3,5,2,7,1,8,6]=>[8,5,4,3,2,7,1,6] [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[4,3,5,2,7,6,8,1]=>[8,5,4,3,2,7,6,1] [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[4,3,5,1,7,2,8,6]=>[8,5,4,3,1,7,2,6] [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[4,1,5,3,7,2,8,6]=>[8,5,4,1,3,7,2,6] [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,1,5,4,7,2,8,6]=>[8,3,1,5,4,7,2,6] [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,5,4,7,3,8,6]=>[2,1,8,5,4,7,3,6] [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,5,4,7,6,8,3]=>[2,1,8,5,4,7,6,3] [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[3,1,5,4,7,6,8,2]=>[8,3,1,5,4,7,6,2] [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[4,1,5,3,7,6,8,2]=>[8,5,4,1,3,7,6,2] [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[4,3,6,5,7,1,8,2]=>[8,7,4,3,6,5,1,2] [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1]=>[8,7,4,3,6,5,2,1] [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[5,3,6,4,7,1,8,2]=>[8,7,6,5,3,4,1,2] [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[5,1,6,3,7,4,8,2]=>[8,7,6,5,1,3,4,2] [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[4,1,6,3,7,5,8,2]=>[8,7,4,1,6,3,5,2] [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[3,1,6,4,7,5,8,2]=>[8,3,1,7,6,4,5,2] [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,6,4,7,5,8,3]=>[2,1,8,7,6,4,5,3] [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,6,4,7,3,8,5]=>[2,1,8,7,6,4,3,5] [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[3,1,6,4,7,2,8,5]=>[8,3,1,7,6,4,2,5] [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[4,1,6,3,7,2,8,5]=>[8,7,4,1,6,3,2,5] [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[5,1,6,3,7,2,8,4]=>[8,7,6,5,1,3,2,4] [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[5,3,6,2,7,4,8,1]=>[8,7,6,5,3,2,4,1] [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[5,2,6,3,7,4,8,1]=>[8,7,6,5,2,3,4,1] [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[5,2,6,3,7,1,8,4]=>[8,7,6,5,2,3,1,4] [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[5,2,6,1,7,3,8,4]=>[8,7,6,5,2,1,3,4] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>[8,7,6,5,1,2,3,4] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,1,6,2,7,3,8,5]=>[8,7,4,1,6,2,3,5] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,1,6,2,7,4,8,5]=>[8,3,1,7,6,2,4,5] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,6,3,7,4,8,5]=>[2,1,8,7,6,3,4,5] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,5,3,7,4,8,6]=>[2,1,8,5,3,7,4,6] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,1,5,2,7,4,8,6]=>[8,3,1,5,2,7,4,6] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,1,5,2,7,3,8,6]=>[8,5,4,1,2,7,3,6] [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[4,2,5,1,7,3,8,6]=>[8,5,4,2,1,7,3,6] [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[4,2,6,1,7,3,8,5]=>[8,7,4,2,6,1,3,5] [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[4,2,6,3,7,1,8,5]=>[8,7,4,2,6,3,1,5] [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[4,2,6,3,7,5,8,1]=>[8,7,4,2,6,3,5,1] [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[3,2,6,4,7,5,8,1]=>[8,3,2,7,6,4,5,1] [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[3,2,6,4,7,1,8,5]=>[8,3,2,7,6,4,1,5] [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[3,2,6,1,7,4,8,5]=>[8,3,2,7,6,1,4,5] [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[3,2,5,1,7,4,8,6]=>[8,3,2,5,1,7,4,6] [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[3,2,4,1,7,5,8,6]=>[4,3,2,1,8,7,5,6] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,1,4,2,7,5,8,6]=>[4,3,1,2,8,7,5,6] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,5,8,6]=>[2,1,4,3,8,7,5,6] [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,8,5]=>[2,1,4,3,8,7,6,5] [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[3,1,4,2,7,6,8,5]=>[4,3,1,2,8,7,6,5] [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>[4,3,2,1,8,7,6,5] [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,2,5,1,7,6,8,4]=>[8,3,2,5,1,7,6,4] [(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[3,2,6,1,7,5,8,4]=>[8,3,2,7,6,1,5,4] [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]=>[8,3,2,7,6,5,4,1] [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[4,2,6,1,7,5,8,3]=>[8,7,4,2,6,1,5,3] [(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[4,2,5,1,7,6,8,3]=>[8,5,4,2,1,7,6,3] [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[4,1,5,2,7,6,8,3]=>[8,5,4,1,2,7,6,3] [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[3,1,5,2,7,6,8,4]=>[8,3,1,5,2,7,6,4] [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,5,3,7,6,8,4]=>[2,1,8,5,3,7,6,4] [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,6,3,7,5,8,4]=>[2,1,8,7,6,3,5,4] [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[3,1,6,2,7,5,8,4]=>[8,3,1,7,6,2,5,4] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[4,1,6,2,7,5,8,3]=>[8,7,4,1,6,2,5,3] [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[5,1,6,2,7,4,8,3]=>[8,7,6,5,1,2,4,3] [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[5,2,6,1,7,4,8,3]=>[8,7,6,5,2,1,4,3] [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[5,4,6,2,7,1,8,3]=>[8,7,6,5,4,2,1,3] [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[5,1,6,4,7,2,8,3]=>[8,7,6,5,1,4,2,3] [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[3,1,6,5,7,2,8,4]=>[8,3,1,7,6,5,2,4] [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[2,1,8,7,6,5,4,3] [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[8,3,1,7,6,5,4,2] [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[8,7,4,1,6,5,3,2] [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]=>[8,7,6,5,1,4,3,2] [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]=>[8,7,6,5,4,3,2,1] [(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]=>[2,1,4,3,6,5,8,7,10,9] [(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]=>[2,1,4,3,6,5,8,7,10,9,12,11]
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
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$(des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi),$$
where
• $des$ is the number of descents, St000021The number of descents of a permutation.,
• $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
• $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
• $Ebot$ is the sum of the excedance bottoms,
• $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
• $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
• $Res$ is the sum of the (right) embracing numbers,
• $Ine$ is the sum of the side numbers.