Identifier
Values
[1] => [1,0] => [2,1] => 1
[1,2] => [1,0,1,0] => [3,1,2] => 2
[2,1] => [1,1,0,0] => [2,3,1] => 2
[1,2,3] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,3,2] => [1,0,1,1,0,0] => [3,1,4,2] => 6
[2,1,3] => [1,1,0,0,1,0] => [2,4,1,3] => 6
[2,3,1] => [1,1,0,1,0,0] => [4,3,1,2] => 6
[3,1,2] => [1,1,1,0,0,0] => [2,3,4,1] => 6
[3,2,1] => [1,1,1,0,0,0] => [2,3,4,1] => 6
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 24
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 24
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 24
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 24
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 24
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 24
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 24
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 24
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 24
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 20
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 24
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 24
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 24
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 24
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 24
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 24
[3,4,1,2] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 24
[3,4,2,1] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 24
[4,1,2,3] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 24
[4,1,3,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 24
[4,2,1,3] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 24
[4,2,3,1] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 24
[4,3,1,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 24
[4,3,2,1] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 24
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 120
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 120
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 120
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 120
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 120
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 120
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 120
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 120
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 120
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 90
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 120
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 120
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 120
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 120
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 120
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 120
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 120
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 120
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 120
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 120
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 120
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 120
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 120
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 120
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 120
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 120
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 120
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 120
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 120
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 120
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => 120
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 120
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 90
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 40
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 90
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 90
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 120
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 120
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 120
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 120
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 40
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 40
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 120
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 120
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 120
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 120
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 120
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 120
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 120
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 120
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 120
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 90
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 120
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 120
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 120
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 120
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 120
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 90
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 120
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 120
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 120
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 40
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 120
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 40
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 90
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 90
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 120
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 120
>>> Load all 153 entries. <<<
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 120
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 120
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 120
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 120
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 120
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 120
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 120
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 120
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 120
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 120
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 120
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 120
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 120
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 120
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 120
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 120
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 120
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 120
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 120
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 120
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 120
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 120
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 120
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 120
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 120
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 120
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 120
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 120
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 120
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Description
The size of the conjugacy class of a permutation.
Two permutations are conjugate if and only if they have the same cycle type, this statistic is then computed as described in St000182The number of permutations whose cycle type is the given integer partition..
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.