Identifier
Values
[(1,2)] => [2,1] => [1,2] => [1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [3,4,1,2] => [3,1,2] => 0
[(1,3),(2,4)] => [3,4,1,2] => [2,1,4,3] => [2,1,3] => 0
[(1,4),(2,3)] => [3,4,2,1] => [2,1,3,4] => [2,1,3] => 0
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => [5,3,4,1,2] => 0
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,3,6,5,1,2] => [4,3,5,1,2] => 0
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,3,5,6,1,2] => [4,3,5,1,2] => 0
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [4,2,5,1,6,3] => [4,2,5,1,3] => 0
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [4,2,5,1,3,6] => [4,2,5,1,3] => 0
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [3,2,1,5,4,6] => [3,2,1,5,4] => 1
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [3,2,1,5,6,4] => [3,2,1,5,4] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,5,4] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [4,2,6,1,5,3] => [4,2,1,5,3] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [5,6,2,1,4,3] => [5,2,1,4,3] => 1
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [5,6,2,1,3,4] => [5,2,1,3,4] => 1
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [4,2,6,1,3,5] => [4,2,1,3,5] => 1
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [3,2,1,6,4,5] => [3,2,1,4,5] => 1
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [3,2,1,4,6,5] => [3,2,1,4,5] => 1
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [3,2,1,4,5,6] => [3,2,1,4,5] => 1
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => 0
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [6,5,8,7,3,4,1,2] => [6,5,7,3,4,1,2] => 0
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => 0
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [6,4,7,3,8,5,1,2] => [6,4,7,3,5,1,2] => 0
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [6,4,7,3,5,8,1,2] => [6,4,7,3,5,1,2] => 0
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [6,4,7,2,5,1,8,3] => [6,4,7,2,5,1,3] => 0
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [6,4,7,2,5,1,3,8] => [6,4,7,2,5,1,3] => 0
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [5,3,2,6,1,8,4,7] => [5,3,2,6,1,4,7] => 1
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [5,3,2,6,1,4,8,7] => [5,3,2,6,1,4,7] => 1
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [5,3,2,6,1,4,7,8] => [5,3,2,6,1,4,7] => 1
[(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => [7,8,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => 2
[(1,5),(2,8),(3,7),(4,6)] => [5,6,7,8,1,4,3,2] => [4,3,2,1,8,5,6,7] => [4,3,2,1,5,6,7] => 2
[(1,6),(2,8),(3,7),(4,5)] => [5,6,7,8,4,1,3,2] => [4,3,2,1,5,8,6,7] => [4,3,2,1,5,6,7] => 2
[(1,7),(2,8),(3,6),(4,5)] => [5,6,7,8,4,3,1,2] => [4,3,2,1,5,6,8,7] => [4,3,2,1,5,6,7] => 2
[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7] => 2
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,4,1,2] => [9,7,8,5,6,3,4,1,2] => 0
[(1,10),(2,3),(4,5),(6,7),(8,9)] => [3,5,2,7,4,9,6,10,8,1] => [8,6,9,4,7,2,5,1,3,10] => [8,6,9,4,7,2,5,1,3] => 0
[(1,10),(2,9),(3,4),(5,6),(7,8)] => [4,6,8,3,9,5,10,7,2,1] => [7,5,3,8,2,6,1,4,9,10] => [7,5,3,8,2,6,1,4,9] => 1
[(1,10),(2,9),(3,8),(4,5),(6,7)] => [5,7,8,9,4,10,6,3,2,1] => [6,4,3,2,7,1,5,8,9,10] => [6,4,3,2,7,1,5,8,9] => 2
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [6,7,8,9,10,5,4,3,2,1] => [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => 3
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 double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
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
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$