Identifier
Values
{{1}} => [1] => [1,0] => [2,1] => 0
{{1,2}} => [2,1] => [1,1,0,0] => [2,3,1] => 0
{{1},{2}} => [1,2] => [1,0,1,0] => [3,1,2] => 0
{{1,2,3}} => [2,3,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
{{1,2},{3}} => [2,1,3] => [1,1,0,0,1,0] => [2,4,1,3] => 0
{{1,3},{2}} => [3,2,1] => [1,1,1,0,0,0] => [2,3,4,1] => 0
{{1},{2,3}} => [1,3,2] => [1,0,1,1,0,0] => [3,1,4,2] => 0
{{1},{2},{3}} => [1,2,3] => [1,0,1,0,1,0] => [4,1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 1
{{1,2,3},{4}} => [2,3,1,4] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
{{1,2,4},{3}} => [2,4,3,1] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 1
{{1,2},{3,4}} => [2,1,4,3] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 1
{{1,3},{2,4}} => [3,4,1,2] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 2
{{1,3},{2},{4}} => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
{{1,4},{2,3}} => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
{{1},{2,3,4}} => [1,3,4,2] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
{{1},{2,3},{4}} => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
{{1},{2,4},{3}} => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
{{1},{2},{3,4}} => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 0
{{1,2,3,4},{5}} => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 1
{{1,2,3,5},{4}} => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 1
{{1,2,3},{4,5}} => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 1
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => 1
{{1,2,4,5},{3}} => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 1
{{1,2,5},{3,4}} => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 1
{{1,2},{3,4,5}} => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 1
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 1
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 0
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
{{1,3,4,5},{2}} => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 1
{{1,3,4},{2,5}} => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 2
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 1
{{1,3,5},{2,4}} => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 3
{{1,3},{2,4,5}} => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 2
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 1
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 0
{{1,4,5},{2,3}} => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 1
{{1,4},{2,3,5}} => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 2
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
{{1,5},{2,3,4}} => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 1
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 0
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 1
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 3
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 2
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 1
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 0
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 0
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Description
The number of nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see St000155The number of exceedances (also excedences) of a permutation..
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.