Identifier
Values
{{1}} => [1] => [1,0,1,0] => [3,1,2] => 0
{{1,2}} => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 0
{{1},{2}} => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 1
{{1,2,3}} => [3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
{{1,2},{3}} => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
{{1,3},{2}} => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
{{1},{2,3}} => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
{{1},{2},{3}} => [1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
{{1,2,3},{4}} => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 0
{{1,2,4},{3}} => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 0
{{1,2},{3,4}} => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
{{1,2},{3},{4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 2
{{1,3,4},{2}} => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 0
{{1,3},{2,4}} => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
{{1,3},{2},{4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 2
{{1,4},{2,3}} => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
{{1},{2,3,4}} => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 0
{{1},{2,3},{4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 2
{{1,4},{2},{3}} => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 2
{{1},{2,4},{3}} => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 2
{{1},{2},{3,4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 2
{{1,2,3},{4,5}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,2,3},{4},{5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1,2,4},{3,5}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,2,4},{3},{5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1,2,5},{3,4}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,2},{3,4,5}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,2},{3,4},{5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,2,5},{3},{4}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1,2},{3,5},{4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,2},{3},{4,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,3,4},{2,5}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,3,4},{2},{5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1,3,5},{2,4}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,3},{2,4,5}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,3},{2,4},{5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,3,5},{2},{4}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1,3},{2,5},{4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,3},{2},{4,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,4,5},{2,3}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,4},{2,3,5}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1,4},{2,3},{5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,5},{2,3,4}} => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
{{1},{2,3,4},{5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1,5},{2,3},{4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1},{2,3,5},{4}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1},{2,3},{4,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,4,5},{2},{3}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1,4},{2,5},{3}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,4},{2},{3,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,5},{2,4},{3}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1},{2,4,5},{3}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1},{2,4},{3,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1,5},{2},{3,4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1},{2,5},{3,4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
{{1},{2},{3,4,5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
{{1,2,3},{4,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,3},{4,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,3},{4},{5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,4},{3,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,4},{3,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,4},{3},{5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,5},{3,4},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2},{3,4,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,6},{3,4},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2},{3,4,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,5},{3,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,5},{3},{4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,6},{3,5},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2},{3,5,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2,6},{3},{4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,2},{3},{4,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3,4},{2,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3,4},{2,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3,4},{2},{5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3,5},{2,4},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3},{2,4,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3,6},{2,4},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3},{2,4,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3,5},{2,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3,5},{2},{4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3,6},{2,5},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3},{2,5,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3,6},{2},{4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,3},{2},{4,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,4,5},{2,3},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,4},{2,3,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,4,6},{2,3},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,4},{2,3,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,5},{2,3,4},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,6},{2,3,4},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,3,4},{5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,5,6},{2,3},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,5},{2,3,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,6},{2,3,5},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,3,5},{4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,3,6},{4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,3},{4,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,4,5},{2,6},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,4,5},{2},{3,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,4,6},{2,5},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
>>> Load all 116 entries. <<<
{{1,4},{2,5,6},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,4,6},{2},{3,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,4},{2},{3,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,5,6},{2,4},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,5},{2,4,6},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,6},{2,4,5},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,4,5},{3,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,4,6},{3,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,4},{3,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,5,6},{2},{3,4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,5},{2},{3,4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,5,6},{3,4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,5},{3,4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1,6},{2},{3,4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
{{1},{2,6},{3,4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
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Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the second entry is St001557The number of inversions of the second entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
shape
Description
Sends a set partition to the integer partition obtained by the sizes of the blocks.