Identifier
Values
[(1,2)] => [2,1] => [1] => [1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [2,1,3] => [3,2,1] => 1
[(1,3),(2,4)] => [3,4,1,2] => [3,1,2] => [3,1,2] => 1
[(1,4),(2,3)] => [4,3,2,1] => [3,2,1] => [1,3,2] => 1
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,4,1,2,5] => [4,5,2,3,1] => 2
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [4,3,2,1,5] => [5,4,3,2,1] => 2
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [5,3,2,1,4] => [5,4,3,1,2] => 2
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [3,2,5,4,1] => [1,3,2,5,4] => 2
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,5,2,3,1] => [1,4,5,2,3] => 1
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [5,4,2,1,3] => [5,4,1,3,2] => 2
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,5,1,2,4] => [4,5,2,1,3] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,4] => [3,2,5,1,4] => 2
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,5,4,3] => [3,2,1,5,4] => 2
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [3,1,5,4,2] => [3,1,2,5,4] => 2
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,5,1,3,2] => [4,1,5,2,3] => 2
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [5,4,3,1,2] => [5,1,4,3,2] => 2
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 2
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,1] => [1,3,2,5,4,7,6] => 3
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [4,5,2,3,7,6,1] => [1,4,5,2,3,7,6] => 2
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [4,6,2,7,3,5,1] => [1,4,6,2,7,3,5] => 2
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [3,2,6,7,4,5,1] => [1,3,2,6,7,4,5] => 2
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [3,2,7,6,5,4,1] => [1,3,2,7,6,5,4] => 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 non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
Map
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.