Identifier
Values
[(1,2)] => [2,1] => [2,1] => [2,1] => 1
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 1
[(1,3),(2,4)] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[(1,4),(2,3)] => [3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,4,6,1,3,5] => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,3,2,1,6,5] => [4,3,2,6,1,5] => 2
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,3,2,1,6,5] => [4,3,2,6,1,5] => 2
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [5,6,3,4,1,2] => [5,3,1,6,4,2] => 2
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 1
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [5,3,1,6,4,2] => 2
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,5,4,3] => [6,5,2,4,1,3] => 2
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,5,4,3] => [6,5,2,4,1,3] => 2
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 1
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 1
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [7,5,3,8,2,6,1,4] => [3,5,2,7,1,8,6,4] => 3
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [7,5,3,8,2,6,1,4] => [3,5,2,7,1,8,6,4] => 3
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,6),(3,8),(4,7)] => [5,6,7,8,1,2,4,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,5),(3,8),(4,7)] => [5,6,7,8,2,1,4,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,5),(3,8),(4,6)] => [5,6,7,8,2,4,1,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,5),(3,7),(4,6)] => [5,6,7,8,2,4,3,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,6),(3,7),(4,5)] => [5,6,7,8,4,2,3,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,6),(3,8),(4,5)] => [5,6,7,8,4,2,1,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,7),(3,8),(4,5)] => [5,6,7,8,4,1,2,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,7),(3,8),(4,6)] => [5,6,7,8,1,4,2,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,8),(3,7),(4,6)] => [5,6,7,8,1,4,3,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,8),(3,7),(4,5)] => [5,6,7,8,4,1,3,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,8),(3,6),(4,5)] => [5,6,7,8,4,3,1,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 1
[(1,10),(2,9),(3,8),(4,6),(5,7)] => [6,7,8,9,10,4,5,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,6),(2,10),(3,9),(4,7),(5,8)] => [6,7,8,9,10,1,4,5,3,2] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,6),(2,9),(3,10),(4,7),(5,8)] => [6,7,8,9,10,1,4,5,2,3] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,10),(2,8),(3,9),(4,6),(5,7)] => [6,7,8,9,10,4,5,2,3,1] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,9),(2,8),(3,7),(4,6),(5,10)] => [6,7,8,9,10,4,3,2,1,5] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,10),(2,9),(3,6),(4,7),(5,8)] => [6,7,8,9,10,3,4,5,2,1] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,9),(2,10),(3,6),(4,7),(5,8)] => [6,7,8,9,10,3,4,5,1,2] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,6),(2,10),(3,7),(4,8),(5,9)] => [6,7,8,9,10,1,3,4,5,2] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,8),(2,9),(3,6),(4,7),(5,10)] => [6,7,8,9,10,3,4,1,2,5] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,9),(2,8),(3,6),(4,7),(5,10)] => [6,7,8,9,10,3,4,2,1,5] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,6),(2,7),(3,8),(4,9),(5,10)] => [6,7,8,9,10,1,2,3,4,5] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,9),(2,6),(3,7),(4,8),(5,10)] => [6,7,8,9,10,2,3,4,1,5] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,10),(2,6),(3,7),(4,8),(5,9)] => [6,7,8,9,10,2,3,4,5,1] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,6),(2,10),(3,9),(4,8),(5,7)] => [6,7,8,9,10,1,5,4,3,2] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)] => [7,8,9,10,11,12,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 1
[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)] => [7,8,9,10,11,12,1,2,3,4,5,6] => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with St000366The number of double descents of a permutation. is studied in [3], where left outer peaks are called exterior peaks.
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.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.