Your data matches 28 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000711
(load all 2 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => 0
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => 0
[2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => 0
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 2
[1,4,5,3,2] => [1,4,5,2,3] => 2
Description
The number of big exceedences of a permutation. A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$. This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St000052
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[2,1] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[2,1,3] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[3,1,2] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,4,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,3,4] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[2,1,4,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,3,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[3,1,2,4] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[3,1,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,2,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[3,4,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[4,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[4,2,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[4,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[4,3,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000710
(load all 5 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 94%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => 0
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 1
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 0
[2,1,4,3] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,3,5,2,4] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,3,5,2,4] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,3,4,5,2,6,7] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 0
[1,3,4,5,2,7,6] => [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => ? = 0
[1,3,4,5,6,2,7] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 0
[1,3,4,5,6,7,2] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 0
[1,3,4,5,7,2,6] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 1
[1,3,4,5,7,6,2] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 1
[1,3,4,6,2,5,7] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 1
[1,3,4,6,2,7,5] => [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => ? = 1
[1,3,4,6,5,2,7] => [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => ? = 1
[1,3,4,6,5,7,2] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 1
[1,3,4,6,7,2,5] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 2
[1,3,4,6,7,5,2] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 2
[1,3,4,7,2,5,6] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 1
[1,3,4,7,2,6,5] => [1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => ? = 1
[1,3,4,7,5,2,6] => [1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => ? = 1
[1,3,4,7,5,6,2] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 1
[1,3,4,7,6,2,5] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 1
[1,3,4,7,6,5,2] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 1
[1,3,5,6,2,4,7] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 2
[1,3,5,6,2,7,4] => [1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => ? = 2
[1,3,5,6,4,2,7] => [1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => ? = 2
[1,3,5,6,4,7,2] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 2
[1,3,5,6,7,2,4] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 3
[1,3,5,6,7,4,2] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 3
[1,3,5,7,2,4,6] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2
[1,3,5,7,2,6,4] => [1,3,5,7,2,6,4] => [1,5,2,7,3,6,4] => ? = 2
[1,3,5,7,4,2,6] => [1,3,5,7,2,6,4] => [1,5,2,7,3,6,4] => ? = 2
[1,3,5,7,4,6,2] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2
[1,3,5,7,6,2,4] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2
[1,3,5,7,6,4,2] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2
[1,3,6,7,2,4,5] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 2
[1,3,6,7,2,5,4] => [1,3,6,7,2,5,4] => [1,5,2,7,6,3,4] => ? = 2
[1,3,6,7,4,2,5] => [1,3,6,7,2,5,4] => [1,5,2,7,6,3,4] => ? = 2
[1,3,6,7,4,5,2] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 2
[1,3,6,7,5,2,4] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 2
[1,3,6,7,5,4,2] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 2
[1,4,5,2,6,7,3] => [1,4,5,2,6,7,3] => [1,4,7,2,3,5,6] => ? = 2
[1,4,5,3,2,6,7] => [1,4,5,2,6,7,3] => [1,4,7,2,3,5,6] => ? = 2
[1,4,5,6,2,3,7] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 3
[1,4,5,6,2,7,3] => [1,4,5,6,2,7,3] => [1,5,7,2,3,4,6] => ? = 3
[1,4,5,6,3,2,7] => [1,4,5,6,2,7,3] => [1,5,7,2,3,4,6] => ? = 3
[1,4,5,6,3,7,2] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 3
[1,4,5,6,7,2,3] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 4
[1,4,5,6,7,3,2] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 4
[1,4,5,7,2,3,6] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 3
[1,4,5,7,2,6,3] => [1,4,5,7,2,6,3] => [1,5,7,2,3,6,4] => ? = 3
[1,4,5,7,3,2,6] => [1,4,5,7,2,6,3] => [1,5,7,2,3,6,4] => ? = 3
[1,4,5,7,3,6,2] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 3
[1,4,5,7,6,2,3] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 3
[1,4,5,7,6,3,2] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 3
Description
The number of big deficiencies of a permutation. A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$. This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.
Matching statistic: St000647
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 81%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0
[2,1,4,3] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 2
[1,3,4,5,2,6,7] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 0
[1,3,4,5,2,7,6] => [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => [1,5,4,3,2,7,6] => ? = 0
[1,3,4,5,6,2,7] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 0
[1,3,4,5,7,2,6] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => [1,7,5,4,3,2,6] => ? = 1
[1,3,4,5,7,6,2] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => [1,7,5,4,3,2,6] => ? = 1
[1,3,4,6,2,5,7] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => [1,6,4,3,2,5,7] => ? = 1
[1,3,4,6,2,7,5] => [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => [1,7,6,4,3,2,5] => ? = 1
[1,3,4,6,5,2,7] => [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => [1,7,6,4,3,2,5] => ? = 1
[1,3,4,6,5,7,2] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => [1,6,4,3,2,5,7] => ? = 1
[1,3,4,6,7,2,5] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => [1,6,4,3,2,7,5] => ? = 2
[1,3,4,6,7,5,2] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => [1,6,4,3,2,7,5] => ? = 2
[1,3,4,7,2,5,6] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 1
[1,3,4,7,2,6,5] => [1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => [1,6,7,4,3,2,5] => ? = 1
[1,3,4,7,5,2,6] => [1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => [1,6,7,4,3,2,5] => ? = 1
[1,3,4,7,5,6,2] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 1
[1,3,4,7,6,2,5] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 1
[1,3,4,7,6,5,2] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 1
[1,3,5,2,4,6,7] => [1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => [1,5,3,2,4,6,7] => ? = 1
[1,3,5,2,4,7,6] => [1,3,5,2,4,7,6] => [1,4,2,5,3,7,6] => [1,5,3,2,4,7,6] => ? = 1
[1,3,5,2,6,4,7] => [1,3,5,2,6,4,7] => [1,4,2,6,3,5,7] => [1,6,5,3,2,4,7] => ? = 1
[1,3,5,2,6,7,4] => [1,3,5,2,6,7,4] => [1,4,2,7,3,5,6] => [1,7,6,5,3,2,4] => ? = 1
[1,3,5,2,7,4,6] => [1,3,5,2,7,4,6] => [1,4,2,6,3,7,5] => [1,7,5,3,2,4,6] => ? = 2
[1,3,5,2,7,6,4] => [1,3,5,2,7,4,6] => [1,4,2,6,3,7,5] => [1,7,5,3,2,4,6] => ? = 2
[1,3,5,4,2,6,7] => [1,3,5,2,6,7,4] => [1,4,2,7,3,5,6] => [1,7,6,5,3,2,4] => ? = 1
[1,3,5,4,2,7,6] => [1,3,5,2,7,4,6] => [1,4,2,6,3,7,5] => [1,7,5,3,2,4,6] => ? = 2
[1,3,5,4,6,2,7] => [1,3,5,2,7,4,6] => [1,4,2,6,3,7,5] => [1,7,5,3,2,4,6] => ? = 2
[1,3,5,4,6,7,2] => [1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => [1,5,3,2,4,6,7] => ? = 1
[1,3,5,4,7,2,6] => [1,3,5,2,6,4,7] => [1,4,2,6,3,5,7] => [1,6,5,3,2,4,7] => ? = 1
[1,3,5,4,7,6,2] => [1,3,5,2,4,7,6] => [1,4,2,5,3,7,6] => [1,5,3,2,4,7,6] => ? = 1
[1,3,5,6,2,4,7] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => [1,5,3,2,6,4,7] => ? = 2
[1,3,5,6,2,7,4] => [1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => [1,5,3,2,7,6,4] => ? = 2
[1,3,5,6,4,2,7] => [1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => [1,5,3,2,7,6,4] => ? = 2
[1,3,5,6,4,7,2] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => [1,5,3,2,6,4,7] => ? = 2
[1,3,5,6,7,2,4] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => [1,7,5,3,2,6,4] => ? = 3
[1,3,5,6,7,4,2] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => [1,7,5,3,2,6,4] => ? = 3
[1,3,5,7,2,4,6] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => [1,5,3,2,7,4,6] => ? = 2
[1,3,5,7,2,6,4] => [1,3,5,7,2,6,4] => [1,5,2,7,3,6,4] => [1,5,3,2,6,7,4] => ? = 2
[1,3,5,7,4,2,6] => [1,3,5,7,2,6,4] => [1,5,2,7,3,6,4] => [1,5,3,2,6,7,4] => ? = 2
[1,3,5,7,4,6,2] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => [1,5,3,2,7,4,6] => ? = 2
[1,3,5,7,6,2,4] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => [1,5,3,2,7,4,6] => ? = 2
[1,3,5,7,6,4,2] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => [1,5,3,2,7,4,6] => ? = 2
[1,3,6,2,4,5,7] => [1,3,6,2,4,5,7] => [1,4,2,5,6,3,7] => [1,6,3,2,4,5,7] => ? = 1
[1,3,6,2,4,7,5] => [1,3,6,2,4,7,5] => [1,4,2,5,7,3,6] => [1,7,6,3,2,4,5] => ? = 1
[1,3,6,2,5,4,7] => [1,3,6,2,5,4,7] => [1,4,2,6,5,3,7] => [1,5,6,3,2,4,7] => ? = 1
[1,3,6,2,5,7,4] => [1,3,6,2,5,7,4] => [1,4,2,7,5,3,6] => [1,5,7,6,3,2,4] => ? = 1
[1,3,6,2,7,4,5] => [1,3,6,2,7,4,5] => [1,4,2,6,7,3,5] => [1,6,3,2,4,7,5] => ? = 2
[1,3,6,2,7,5,4] => [1,3,6,2,7,4,5] => [1,4,2,6,7,3,5] => [1,6,3,2,4,7,5] => ? = 2
[1,3,6,4,2,5,7] => [1,3,6,2,5,7,4] => [1,4,2,7,5,3,6] => [1,5,7,6,3,2,4] => ? = 1
[1,3,6,4,2,7,5] => [1,3,6,2,7,4,5] => [1,4,2,6,7,3,5] => [1,6,3,2,4,7,5] => ? = 2
[1,3,6,4,5,2,7] => [1,3,6,2,7,4,5] => [1,4,2,6,7,3,5] => [1,6,3,2,4,7,5] => ? = 2
Description
The number of big descents of a permutation. For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$. The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1]. For the number of small descents, see [[St000214]].
Matching statistic: St000703
(load all 4 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1,3] => 1 = 0 + 1
[2,1,3] => [1,3,2] => [2,1,3] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 1 = 0 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,3,4] => 1 = 0 + 1
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,3,4] => 1 = 0 + 1
[2,1,4,3] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 1 = 0 + 1
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => 1 = 0 + 1
[3,1,4,2] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => 1 = 0 + 1
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => 1 = 0 + 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => 1 = 0 + 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,5,3,4,1] => 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,3,4,5] => 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,1,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,5,3,1,4] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,4,1,3,5] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,4,1,3,5] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,4,5,3,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,3,4] => 3 = 2 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,3,4] => 3 = 2 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => ? = 0 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => ? = 0 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,4,6,5,1,7] => ? = 0 + 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [2,3,4,7,5,6,1] => ? = 0 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => ? = 1 + 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => ? = 1 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => ? = 1 + 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [2,3,4,6,1,5,7] => ? = 1 + 1
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [2,3,4,6,1,5,7] => ? = 1 + 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => ? = 1 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,3,5,4,6,7,1] => ? = 0 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [2,3,5,4,6,1,7] => ? = 0 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,3,5,4,7,6,1] => ? = 0 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [2,3,5,4,1,6,7] => ? = 0 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [2,3,5,4,7,1,6] => ? = 1 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,3,5,4,7,1,6] => ? = 1 + 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [2,3,6,4,5,7,1] => ? = 0 + 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [2,3,6,4,5,1,7] => ? = 0 + 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [2,3,7,4,5,6,1] => ? = 0 + 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [2,3,7,4,5,1,6] => ? = 1 + 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [2,3,7,4,5,1,6] => ? = 1 + 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [2,3,6,4,7,5,1] => ? = 1 + 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [2,3,6,4,1,5,7] => ? = 1 + 1
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [2,3,6,4,1,5,7] => ? = 1 + 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [2,3,6,4,7,5,1] => ? = 1 + 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [2,3,7,4,1,5,6] => ? = 2 + 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [2,3,7,4,1,5,6] => ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => ? = 1 + 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [2,3,6,4,1,7,5] => ? = 1 + 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [2,3,6,4,1,7,5] => ? = 1 + 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => ? = 1 + 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => ? = 1 + 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => ? = 1 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [2,3,5,6,4,7,1] => ? = 1 + 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [2,3,5,6,4,1,7] => ? = 1 + 1
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [2,3,5,7,4,6,1] => ? = 1 + 1
[1,2,5,3,6,7,4] => [1,2,5,3,6,7,4] => [2,3,5,1,4,6,7] => ? = 1 + 1
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [2,3,5,7,4,1,6] => ? = 2 + 1
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [2,3,5,7,4,1,6] => ? = 2 + 1
[1,2,5,4,3,6,7] => [1,2,5,3,6,7,4] => [2,3,5,1,4,6,7] => ? = 1 + 1
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [2,3,5,7,4,1,6] => ? = 2 + 1
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [2,3,5,7,4,1,6] => ? = 2 + 1
[1,2,5,4,6,7,3] => [1,2,5,3,4,6,7] => [2,3,5,6,4,7,1] => ? = 1 + 1
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [2,3,5,7,4,6,1] => ? = 1 + 1
[1,2,5,4,7,6,3] => [1,2,5,3,4,7,6] => [2,3,5,6,4,1,7] => ? = 1 + 1
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [2,3,6,7,4,5,1] => ? = 2 + 1
[1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => [2,3,6,1,4,5,7] => ? = 2 + 1
[1,2,5,6,4,3,7] => [1,2,5,6,3,7,4] => [2,3,6,1,4,5,7] => ? = 2 + 1
[1,2,5,6,4,7,3] => [1,2,5,6,3,4,7] => [2,3,6,7,4,5,1] => ? = 2 + 1
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [2,3,7,1,4,5,6] => ? = 3 + 1
Description
The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].
Matching statistic: St000375
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [3,2,1] => 0
[2,1,3] => [1,3,2] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,2,3] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 1
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => 1
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => 0
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => 1
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => 0
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => 0
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => 0
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => 1
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => 1
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => 0
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => 0
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => 0
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => 0
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => 0
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => 0
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => 0
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,2,5,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,2,3,1] => 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 0
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 0
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => ? = 0
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => ? = 0
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 0
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [2,3,6,4,5,7,1] => ? = 0
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [2,3,7,4,5,6,1] => ? = 0
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => ? = 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => ? = 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => ? = 0
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => ? = 0
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 0
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [2,4,3,7,5,6,1] => ? = 0
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => ? = 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [2,5,3,4,6,7,1] => ? = 0
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [2,5,3,4,7,6,1] => ? = 0
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [2,6,3,4,5,7,1] => ? = 0
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [2,7,3,4,5,6,1] => ? = 0
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [2,5,3,6,4,7,1] => ? = 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [2,5,3,7,4,6,1] => ? = 1
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [2,5,3,7,4,6,1] => ? = 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [2,5,3,6,4,7,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [2,6,3,7,4,5,1] => ? = 2
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [2,6,3,7,4,5,1] => ? = 2
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => ? = 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [2,5,3,7,6,4,1] => ? = 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [2,5,3,7,6,4,1] => ? = 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => ? = 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => ? = 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => ? = 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [2,4,5,3,6,7,1] => ? = 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [2,4,5,3,7,6,1] => ? = 1
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].
Matching statistic: St001513
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 0
[1,3,2] => [1,3,2] => [3,2,1] => [3,2,1] => 0
[2,1,3] => [1,3,2] => [3,2,1] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [2,3,1] => [3,1,2] => 0
[3,1,2] => [1,2,3] => [2,3,1] => [3,1,2] => 0
[3,2,1] => [1,2,3] => [2,3,1] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 1
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 1
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 0
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 1
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 0
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 0
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 0
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 1
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 1
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 0
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 0
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 0
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [5,1,3,2,4] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [5,1,3,4,2] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => [5,1,4,2,3] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => [5,1,4,2,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => [5,2,1,3,4] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => [5,2,1,4,3] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => [5,2,3,1,4] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => [5,3,1,2,4] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,2,5,1] => [5,3,1,2,4] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => [5,3,4,1,2] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,2,3,1] => [5,3,4,1,2] => 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 0
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 0
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => ? = 0
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => ? = 0
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => [7,1,2,4,3,6,5] => ? = 0
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [2,3,6,4,5,7,1] => [7,1,2,4,5,3,6] => ? = 0
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => ? = 0
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => [7,1,2,4,6,3,5] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => [7,1,2,4,6,3,5] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => [7,1,2,5,3,4,6] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => [7,1,2,5,3,6,4] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => [7,1,2,5,3,6,4] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => [7,1,2,5,3,4,6] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => [7,1,2,5,6,3,4] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => [7,1,2,5,6,3,4] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => [7,1,3,2,4,5,6] => ? = 0
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => [7,1,3,2,4,6,5] => ? = 0
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => [7,1,3,2,5,4,6] => ? = 0
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [2,4,3,7,5,6,1] => [7,1,3,2,5,6,4] => ? = 0
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => [7,1,3,2,6,4,5] => ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => [7,1,3,2,6,4,5] => ? = 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [2,5,3,4,6,7,1] => [7,1,3,4,2,5,6] => ? = 0
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [2,5,3,4,7,6,1] => [7,1,3,4,2,6,5] => ? = 0
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [2,6,3,4,5,7,1] => [7,1,3,4,5,2,6] => ? = 0
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [2,7,3,4,5,6,1] => [7,1,3,4,5,6,2] => ? = 0
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => [7,1,3,4,6,2,5] => ? = 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => [7,1,3,4,6,2,5] => ? = 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [2,5,3,6,4,7,1] => [7,1,3,5,2,4,6] => ? = 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [2,5,3,7,4,6,1] => [7,1,3,5,2,6,4] => ? = 1
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [2,5,3,7,4,6,1] => [7,1,3,5,2,6,4] => ? = 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [2,5,3,6,4,7,1] => [7,1,3,5,2,4,6] => ? = 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [2,6,3,7,4,5,1] => [7,1,3,5,6,2,4] => ? = 2
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [2,6,3,7,4,5,1] => [7,1,3,5,6,2,4] => ? = 2
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [7,1,3,6,2,4,5] => ? = 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [2,5,3,7,6,4,1] => [7,1,3,6,2,5,4] => ? = 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [2,5,3,7,6,4,1] => [7,1,3,6,2,5,4] => ? = 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [7,1,3,6,2,4,5] => ? = 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [7,1,3,6,2,4,5] => ? = 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [7,1,3,6,2,4,5] => ? = 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [2,4,5,3,6,7,1] => [7,1,4,2,3,5,6] => ? = 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [2,4,5,3,7,6,1] => [7,1,4,2,3,6,5] => ? = 1
Description
The number of nested exceedences of a permutation. For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St001685
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 0
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 0
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 0
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 0
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 0
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 0
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [5,4,1,3,2] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [5,4,1,3,2] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [5,3,1,4,2] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [5,3,1,4,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [5,2,4,1,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [5,2,4,1,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [5,2,1,4,3] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => [5,2,1,4,3] => 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 0
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 0
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 0
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => ? = 0
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => ? = 0
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => ? = 0
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => ? = 0
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [7,6,5,3,1,4,2] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [7,6,5,3,1,4,2] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [7,6,5,2,4,3,1] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [7,6,5,2,4,1,3] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [7,6,5,2,4,1,3] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [7,6,5,2,4,3,1] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [7,6,5,2,1,4,3] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [7,6,5,2,1,4,3] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => ? = 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => ? = 0
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [7,6,4,5,3,1,2] => ? = 0
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => ? = 0
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => [7,6,4,5,2,1,3] => ? = 0
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [7,6,4,5,1,3,2] => ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [7,6,4,5,1,3,2] => ? = 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ? = 0
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [7,6,4,3,5,1,2] => ? = 0
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => [7,6,4,3,2,5,1] => ? = 0
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => ? = 0
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [7,6,4,3,1,5,2] => ? = 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [7,6,4,3,1,5,2] => ? = 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => [7,6,4,2,5,3,1] => ? = 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => [7,6,4,2,5,1,3] => ? = 1
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => [7,6,4,2,5,1,3] => ? = 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => [7,6,4,2,5,3,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => [7,6,4,2,1,5,3] => ? = 2
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => [7,6,4,2,1,5,3] => ? = 2
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [7,6,4,1,5,3,2] => ? = 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [7,6,4,1,5,3,2] => ? = 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [7,6,4,1,5,3,2] => ? = 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [7,6,4,1,5,3,2] => ? = 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [7,6,4,1,5,3,2] => ? = 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [7,6,4,1,5,3,2] => ? = 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ? = 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [7,6,3,5,4,1,2] => ? = 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St000021
(load all 2 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1 = 0 + 1
[2,1,3] => [1,3,2] => [2,1,3] => [2,1,3] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 1 = 0 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[2,1,4,3] => [1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1 = 0 + 1
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1 = 0 + 1
[3,1,4,2] => [1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1 = 0 + 1
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 1 = 0 + 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 1 = 0 + 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,1,2,3] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => [5,4,1,2,3] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => [5,4,1,2,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,1,2,4] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,1,2,5] => 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,5,3,4,1] => [3,4,5,1,2] => 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,1,4] => [3,5,4,1,2] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,5,3,1,4] => [3,5,4,1,2] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => [5,1,2,4,3] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,4,1,3,5] => [4,3,1,2,5] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,4,1,3,5] => [4,3,1,2,5] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,4,5,3,1] => [5,1,2,4,3] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,3,4] => [5,4,3,1,2] => 3 = 2 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,3,4] => [5,4,3,1,2] => 3 = 2 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ? = 0 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,1,2,3,4,5,7] => ? = 0 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,7,1,2,3,4,5] => ? = 0 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [5,1,2,3,4,6,7] => ? = 0 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [7,6,1,2,3,4,5] => ? = 1 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [7,6,1,2,3,4,5] => ? = 1 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [5,7,1,2,3,4,6] => ? = 0 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,4,6,5,1,7] => [5,6,1,2,3,4,7] => ? = 0 + 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [2,3,4,7,5,6,1] => [5,6,7,1,2,3,4] => ? = 0 + 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [2,3,4,1,5,6,7] => [4,1,2,3,5,6,7] => ? = 0 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => [5,7,6,1,2,3,4] => ? = 1 + 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => [5,7,6,1,2,3,4] => ? = 1 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,4,6,5] => ? = 1 + 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [2,3,4,6,1,5,7] => [6,5,1,2,3,4,7] => ? = 1 + 1
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [2,3,4,6,1,5,7] => [6,5,1,2,3,4,7] => ? = 1 + 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,4,6,5] => ? = 1 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [2,3,4,7,1,5,6] => [7,6,5,1,2,3,4] => ? = 2 + 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [2,3,4,7,1,5,6] => [7,6,5,1,2,3,4] => ? = 2 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [2,3,4,6,7,1,5] => [6,1,2,3,4,7,5] => ? = 1 + 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [2,3,4,6,1,7,5] => [7,5,1,2,3,4,6] => ? = 1 + 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [2,3,4,6,1,7,5] => [7,5,1,2,3,4,6] => ? = 1 + 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [2,3,4,6,7,1,5] => [6,1,2,3,4,7,5] => ? = 1 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [2,3,4,6,7,1,5] => [6,1,2,3,4,7,5] => ? = 1 + 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,4,6,7,1,5] => [6,1,2,3,4,7,5] => ? = 1 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,3,5,4,6,7,1] => [4,7,1,2,3,5,6] => ? = 0 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [2,3,5,4,6,1,7] => [4,6,1,2,3,5,7] => ? = 0 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,3,5,4,7,6,1] => [4,6,7,1,2,3,5] => ? = 0 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [2,3,5,4,1,6,7] => [4,5,1,2,3,6,7] => ? = 0 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [2,3,5,4,7,1,6] => [4,7,6,1,2,3,5] => ? = 1 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,3,5,4,7,1,6] => [4,7,6,1,2,3,5] => ? = 1 + 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [2,3,6,4,5,7,1] => [4,5,7,1,2,3,6] => ? = 0 + 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [2,3,6,4,5,1,7] => [4,5,6,1,2,3,7] => ? = 0 + 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [2,3,7,4,5,6,1] => [4,5,6,7,1,2,3] => ? = 0 + 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [2,3,1,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 0 + 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [2,3,7,4,5,1,6] => [4,5,7,6,1,2,3] => ? = 1 + 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [2,3,7,4,5,1,6] => [4,5,7,6,1,2,3] => ? = 1 + 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [2,3,6,4,7,5,1] => [4,7,1,2,3,6,5] => ? = 1 + 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [2,3,6,4,1,5,7] => [4,6,5,1,2,3,7] => ? = 1 + 1
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [2,3,6,4,1,5,7] => [4,6,5,1,2,3,7] => ? = 1 + 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [2,3,6,4,7,5,1] => [4,7,1,2,3,6,5] => ? = 1 + 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [2,3,7,4,1,5,6] => [4,7,6,5,1,2,3] => ? = 2 + 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [2,3,7,4,1,5,6] => [4,7,6,5,1,2,3] => ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => [4,6,1,2,3,7,5] => ? = 1 + 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [2,3,6,4,1,7,5] => [4,7,5,1,2,3,6] => ? = 1 + 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [2,3,6,4,1,7,5] => [4,7,5,1,2,3,6] => ? = 1 + 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => [4,6,1,2,3,7,5] => ? = 1 + 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => [4,6,1,2,3,7,5] => ? = 1 + 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => [4,6,1,2,3,7,5] => ? = 1 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [2,3,5,6,4,7,1] => [7,1,2,3,5,4,6] => ? = 1 + 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [2,3,5,6,4,1,7] => [6,1,2,3,5,4,7] => ? = 1 + 1
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000155
(load all 2 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1 = 0 + 1
[2,1,3] => [1,3,2] => [2,1,3] => [2,1,3] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 1 = 0 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[2,1,4,3] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1 = 0 + 1
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1 = 0 + 1
[3,1,4,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1 = 0 + 1
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 1 = 0 + 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 1 = 0 + 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => [4,1,2,5,3] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => [4,1,2,5,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [5,1,3,2,4] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => [4,1,3,2,5] => 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,5,3,4,1] => [5,1,3,4,2] => 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,1,4] => [4,1,3,5,2] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,5,3,1,4] => [4,1,3,5,2] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => [5,1,4,2,3] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,4,1,3,5] => [3,1,4,2,5] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,4,1,3,5] => [3,1,4,2,5] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,4,5,3,1] => [5,1,4,2,3] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,3,4] => [3,1,4,5,2] => 3 = 2 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,3,4] => [3,1,4,5,2] => 3 = 2 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ? = 0 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,1,2,3,4,5,7] => ? = 0 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 0 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [5,1,2,3,4,6,7] => ? = 0 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,1,2,3,4,7,5] => ? = 1 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,1,2,3,4,7,5] => ? = 1 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 0 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,4,6,5,1,7] => [6,1,2,3,5,4,7] => ? = 0 + 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => ? = 0 + 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [2,3,4,1,5,6,7] => [4,1,2,3,5,6,7] => ? = 0 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => [6,1,2,3,5,7,4] => ? = 1 + 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => [6,1,2,3,5,7,4] => ? = 1 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 1 + 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [2,3,4,6,1,5,7] => [5,1,2,3,6,4,7] => ? = 1 + 1
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [2,3,4,6,1,5,7] => [5,1,2,3,6,4,7] => ? = 1 + 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 1 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [2,3,4,7,1,5,6] => [5,1,2,3,6,7,4] => ? = 2 + 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [2,3,4,7,1,5,6] => [5,1,2,3,6,7,4] => ? = 2 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => ? = 1 + 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [2,3,4,6,1,7,5] => [5,1,2,3,7,4,6] => ? = 1 + 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [2,3,4,6,1,7,5] => [5,1,2,3,7,4,6] => ? = 1 + 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => ? = 1 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => ? = 1 + 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => ? = 1 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => ? = 0 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [2,3,5,4,6,1,7] => [6,1,2,4,3,5,7] => ? = 0 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,3,5,4,7,6,1] => [7,1,2,4,3,6,5] => ? = 0 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [2,3,5,4,1,6,7] => [5,1,2,4,3,6,7] => ? = 0 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [2,3,5,4,7,1,6] => [6,1,2,4,3,7,5] => ? = 1 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,3,5,4,7,1,6] => [6,1,2,4,3,7,5] => ? = 1 + 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [2,3,6,4,5,7,1] => [7,1,2,4,5,3,6] => ? = 0 + 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [2,3,6,4,5,1,7] => [6,1,2,4,5,3,7] => ? = 0 + 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => ? = 0 + 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [2,3,1,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 0 + 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [2,3,7,4,5,1,6] => [6,1,2,4,5,7,3] => ? = 1 + 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [2,3,7,4,5,1,6] => [6,1,2,4,5,7,3] => ? = 1 + 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [2,3,6,4,7,5,1] => [7,1,2,4,6,3,5] => ? = 1 + 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [2,3,6,4,1,5,7] => [5,1,2,4,6,3,7] => ? = 1 + 1
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [2,3,6,4,1,5,7] => [5,1,2,4,6,3,7] => ? = 1 + 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [2,3,6,4,7,5,1] => [7,1,2,4,6,3,5] => ? = 1 + 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [2,3,7,4,1,5,6] => [5,1,2,4,6,7,3] => ? = 2 + 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [2,3,7,4,1,5,6] => [5,1,2,4,6,7,3] => ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => [6,1,2,4,7,3,5] => ? = 1 + 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [2,3,6,4,1,7,5] => [5,1,2,4,7,3,6] => ? = 1 + 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [2,3,6,4,1,7,5] => [5,1,2,4,7,3,6] => ? = 1 + 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => [6,1,2,4,7,3,5] => ? = 1 + 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => [6,1,2,4,7,3,5] => ? = 1 + 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [2,3,6,4,7,1,5] => [6,1,2,4,7,3,5] => ? = 1 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [2,3,5,6,4,7,1] => [7,1,2,5,3,4,6] => ? = 1 + 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [2,3,5,6,4,1,7] => [6,1,2,5,3,4,7] => ? = 1 + 1
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001964The interval resolution global dimension of a poset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001768The number of reduced words of a signed permutation.