Identifier
Values
[(1,2)] => [2,1] => [1,2] => 1
[(1,2),(3,4)] => [2,1,4,3] => [3,2,1,4] => 3
[(1,3),(2,4)] => [3,4,1,2] => [4,1,3,2] => 4
[(1,4),(2,3)] => [3,4,2,1] => [4,1,2,3] => 4
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [3,2,5,4,1,6] => 5
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,5,2,3,1,6] => 5
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,5,3,2,1,6] => 5
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [4,6,3,1,5,2] => 6
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [4,6,3,1,2,5] => 6
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [5,6,1,4,2,3] => 6
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [5,6,1,4,3,2] => 6
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [5,6,1,3,4,2] => 6
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [4,6,2,1,5,3] => 6
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [3,2,6,1,5,4] => 6
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [3,2,6,1,4,5] => 6
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [4,6,2,1,3,5] => 6
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [5,6,1,3,2,4] => 6
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [5,6,1,2,4,3] => 6
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [5,6,1,2,3,4] => 6
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Description
The size of the largest block in the direct sum decomposition of a permutation.
A component of a permutation $\pi$ is a set of consecutive numbers $\{a,a+1,\dots, b\}$ such that $a\leq \pi(i) \leq b$ for all $a\leq i\leq b$.
This statistic is the size of the largest component which does not properly contain another component.
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.