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 first part in the decomposition of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is defined to be the smallest $k > 0$ such that $\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$. This statistic is undefined for the empty permutation.
For the number of parts in the decomposition see St000056The decomposition (or block) number of a permutation..
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.