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
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.