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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St001947
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St001947: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 1
[1,2] => 0
[2,1] => 0
[1,1,1] => 2
[1,1,2] => 1
[1,2,1] => 0
[2,1,1] => 1
[1,1,3] => 1
[1,3,1] => 0
[3,1,1] => 1
[1,2,2] => 1
[2,1,2] => 0
[2,2,1] => 1
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,1,1,1] => 3
[1,1,1,2] => 2
[1,1,2,1] => 1
[1,2,1,1] => 1
[2,1,1,1] => 2
[1,1,1,3] => 2
[1,1,3,1] => 1
[1,3,1,1] => 1
[3,1,1,1] => 2
[1,1,1,4] => 2
[1,1,4,1] => 1
[1,4,1,1] => 1
[4,1,1,1] => 2
[1,1,2,2] => 2
[1,2,1,2] => 0
[1,2,2,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 0
[2,2,1,1] => 2
[1,1,2,3] => 1
[1,1,3,2] => 1
[1,2,1,3] => 0
[1,2,3,1] => 0
[1,3,1,2] => 0
[1,3,2,1] => 0
[2,1,1,3] => 1
[2,1,3,1] => 0
[2,3,1,1] => 1
[3,1,1,2] => 1
[3,1,2,1] => 0
Description
The number of ties in a parking function.
This is the number of indices $i$ such that $p_i=p_{i+1}$.
Matching statistic: St000084
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
St000084: Ordered trees ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> 1 = 0 + 1
[1,1] => [[],[]]
=> 2 = 1 + 1
[1,2] => [[[]]]
=> 1 = 0 + 1
[2,1] => [[[]]]
=> 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> 1 = 0 + 1
[2,3,4,5,6,7,1] => ?
=> ? = 0 + 1
[2,3,4,5,7,1,6] => ?
=> ? = 0 + 1
[2,3,4,6,1,7,5] => ?
=> ? = 0 + 1
[2,3,4,7,1,5,6] => ?
=> ? = 0 + 1
[2,3,4,7,6,1,5] => ?
=> ? = 0 + 1
[2,3,5,1,6,7,4] => ?
=> ? = 0 + 1
[2,3,5,1,7,4,6] => ?
=> ? = 0 + 1
[2,3,6,1,4,7,5] => ?
=> ? = 0 + 1
[2,3,6,5,1,7,4] => ?
=> ? = 0 + 1
[2,3,7,1,4,5,6] => ?
=> ? = 0 + 1
[2,3,7,1,6,4,5] => ?
=> ? = 0 + 1
[2,3,7,5,1,4,6] => ?
=> ? = 0 + 1
[2,3,7,5,6,1,4] => ?
=> ? = 0 + 1
[2,3,7,6,1,4,5] => ?
=> ? = 0 + 1
[2,4,1,5,6,7,3] => ?
=> ? = 0 + 1
[2,4,1,5,7,3,6] => ?
=> ? = 0 + 1
[2,4,1,6,3,7,5] => ?
=> ? = 0 + 1
[2,4,1,7,3,5,6] => ?
=> ? = 0 + 1
[2,4,1,7,6,3,5] => ?
=> ? = 0 + 1
[2,5,1,3,6,7,4] => ?
=> ? = 0 + 1
[2,5,1,3,7,4,6] => ?
=> ? = 0 + 1
[2,5,4,1,6,7,3] => ?
=> ? = 0 + 1
[2,5,4,1,7,3,6] => ?
=> ? = 0 + 1
[2,6,1,3,4,7,5] => ?
=> ? = 0 + 1
[2,6,1,5,3,7,4] => ?
=> ? = 0 + 1
[2,6,4,1,3,7,5] => ?
=> ? = 0 + 1
[2,6,4,5,1,7,3] => ?
=> ? = 0 + 1
[2,6,5,1,3,7,4] => ?
=> ? = 0 + 1
[2,6,7,1,3,4,5] => ?
=> ? = 0 + 1
[2,7,1,3,4,5,6] => ?
=> ? = 0 + 1
[2,7,1,3,6,4,5] => ?
=> ? = 0 + 1
[2,7,1,5,3,4,6] => ?
=> ? = 0 + 1
[2,7,1,5,6,3,4] => ?
=> ? = 0 + 1
[2,7,1,6,3,4,5] => ?
=> ? = 0 + 1
[2,7,4,1,3,5,6] => ?
=> ? = 0 + 1
[2,7,4,1,6,3,5] => ?
=> ? = 0 + 1
[2,7,4,5,1,3,6] => ?
=> ? = 0 + 1
[2,7,4,5,6,1,3] => ?
=> ? = 0 + 1
[2,7,4,6,1,3,5] => ?
=> ? = 0 + 1
[2,7,5,1,3,4,6] => ?
=> ? = 0 + 1
[2,7,5,1,6,3,4] => ?
=> ? = 0 + 1
[2,7,6,5,1,3,4] => ?
=> ? = 0 + 1
[3,1,4,5,6,7,2] => ?
=> ? = 0 + 1
[3,1,4,5,7,2,6] => ?
=> ? = 0 + 1
[3,1,4,6,2,7,5] => ?
=> ? = 0 + 1
[3,1,4,7,2,5,6] => ?
=> ? = 0 + 1
[3,1,4,7,6,2,5] => ?
=> ? = 0 + 1
[3,1,5,2,6,7,4] => ?
=> ? = 0 + 1
[3,1,5,2,7,4,6] => ?
=> ? = 0 + 1
[3,1,6,2,4,7,5] => ?
=> ? = 0 + 1
Description
The number of subtrees.
Matching statistic: St000011
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [[],[]]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2] => [[[]]]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [[[]]]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,4,5,6,7,1] => ?
=> ?
=> ? = 0 + 1
[2,3,4,5,7,1,6] => ?
=> ?
=> ? = 0 + 1
[2,3,4,6,1,7,5] => ?
=> ?
=> ? = 0 + 1
[2,3,4,7,1,5,6] => ?
=> ?
=> ? = 0 + 1
[2,3,4,7,6,1,5] => ?
=> ?
=> ? = 0 + 1
[2,3,5,1,6,7,4] => ?
=> ?
=> ? = 0 + 1
[2,3,5,1,7,4,6] => ?
=> ?
=> ? = 0 + 1
[2,3,6,1,4,7,5] => ?
=> ?
=> ? = 0 + 1
[2,3,6,5,1,7,4] => ?
=> ?
=> ? = 0 + 1
[2,3,7,1,4,5,6] => ?
=> ?
=> ? = 0 + 1
[2,3,7,1,6,4,5] => ?
=> ?
=> ? = 0 + 1
[2,3,7,5,1,4,6] => ?
=> ?
=> ? = 0 + 1
[2,3,7,5,6,1,4] => ?
=> ?
=> ? = 0 + 1
[2,3,7,6,1,4,5] => ?
=> ?
=> ? = 0 + 1
[2,4,1,5,6,7,3] => ?
=> ?
=> ? = 0 + 1
[2,4,1,5,7,3,6] => ?
=> ?
=> ? = 0 + 1
[2,4,1,6,3,7,5] => ?
=> ?
=> ? = 0 + 1
[2,4,1,7,3,5,6] => ?
=> ?
=> ? = 0 + 1
[2,4,1,7,6,3,5] => ?
=> ?
=> ? = 0 + 1
[2,5,1,3,6,7,4] => ?
=> ?
=> ? = 0 + 1
[2,5,1,3,7,4,6] => ?
=> ?
=> ? = 0 + 1
[2,5,4,1,6,7,3] => ?
=> ?
=> ? = 0 + 1
[2,5,4,1,7,3,6] => ?
=> ?
=> ? = 0 + 1
[2,6,1,3,4,7,5] => ?
=> ?
=> ? = 0 + 1
[2,6,1,5,3,7,4] => ?
=> ?
=> ? = 0 + 1
[2,6,4,1,3,7,5] => ?
=> ?
=> ? = 0 + 1
[2,6,4,5,1,7,3] => ?
=> ?
=> ? = 0 + 1
[2,6,5,1,3,7,4] => ?
=> ?
=> ? = 0 + 1
[2,6,7,1,3,4,5] => ?
=> ?
=> ? = 0 + 1
[2,7,1,3,4,5,6] => ?
=> ?
=> ? = 0 + 1
[2,7,1,3,6,4,5] => ?
=> ?
=> ? = 0 + 1
[2,7,1,5,3,4,6] => ?
=> ?
=> ? = 0 + 1
[2,7,1,5,6,3,4] => ?
=> ?
=> ? = 0 + 1
[2,7,1,6,3,4,5] => ?
=> ?
=> ? = 0 + 1
[2,7,4,1,3,5,6] => ?
=> ?
=> ? = 0 + 1
[2,7,4,1,6,3,5] => ?
=> ?
=> ? = 0 + 1
[2,7,4,5,1,3,6] => ?
=> ?
=> ? = 0 + 1
[2,7,4,5,6,1,3] => ?
=> ?
=> ? = 0 + 1
[2,7,4,6,1,3,5] => ?
=> ?
=> ? = 0 + 1
[2,7,5,1,3,4,6] => ?
=> ?
=> ? = 0 + 1
[2,7,5,1,6,3,4] => ?
=> ?
=> ? = 0 + 1
[2,7,6,5,1,3,4] => ?
=> ?
=> ? = 0 + 1
[3,1,4,5,6,7,2] => ?
=> ?
=> ? = 0 + 1
[3,1,4,5,7,2,6] => ?
=> ?
=> ? = 0 + 1
[3,1,4,6,2,7,5] => ?
=> ?
=> ? = 0 + 1
[3,1,4,7,2,5,6] => ?
=> ?
=> ? = 0 + 1
[3,1,4,7,6,2,5] => ?
=> ?
=> ? = 0 + 1
[3,1,5,2,6,7,4] => ?
=> ?
=> ? = 0 + 1
[3,1,5,2,7,4,6] => ?
=> ?
=> ? = 0 + 1
[3,1,6,2,4,7,5] => ?
=> ?
=> ? = 0 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000234
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [1,0]
=> [1] => 0
[1,1] => [[],[]]
=> [1,0,1,0]
=> [1,2] => 1
[1,2] => [[[]]]
=> [1,1,0,0]
=> [2,1] => 0
[2,1] => [[[]]]
=> [1,1,0,0]
=> [2,1] => 0
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 0
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 0
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 0
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[2,3,4,5,6,7,1] => ?
=> ?
=> ? => ? = 0
[2,3,4,5,7,1,6] => ?
=> ?
=> ? => ? = 0
[2,3,4,6,1,7,5] => ?
=> ?
=> ? => ? = 0
[2,3,4,7,1,5,6] => ?
=> ?
=> ? => ? = 0
[2,3,4,7,6,1,5] => ?
=> ?
=> ? => ? = 0
[2,3,5,1,6,7,4] => ?
=> ?
=> ? => ? = 0
[2,3,5,1,7,4,6] => ?
=> ?
=> ? => ? = 0
[2,3,6,1,4,7,5] => ?
=> ?
=> ? => ? = 0
[2,3,6,5,1,7,4] => ?
=> ?
=> ? => ? = 0
[2,3,7,1,4,5,6] => ?
=> ?
=> ? => ? = 0
[2,3,7,1,6,4,5] => ?
=> ?
=> ? => ? = 0
[2,3,7,5,1,4,6] => ?
=> ?
=> ? => ? = 0
[2,3,7,5,6,1,4] => ?
=> ?
=> ? => ? = 0
[2,3,7,6,1,4,5] => ?
=> ?
=> ? => ? = 0
[2,4,1,5,6,7,3] => ?
=> ?
=> ? => ? = 0
[2,4,1,5,7,3,6] => ?
=> ?
=> ? => ? = 0
[2,4,1,6,3,7,5] => ?
=> ?
=> ? => ? = 0
[2,4,1,7,3,5,6] => ?
=> ?
=> ? => ? = 0
[2,4,1,7,6,3,5] => ?
=> ?
=> ? => ? = 0
[2,5,1,3,6,7,4] => ?
=> ?
=> ? => ? = 0
[2,5,1,3,7,4,6] => ?
=> ?
=> ? => ? = 0
[2,5,4,1,6,7,3] => ?
=> ?
=> ? => ? = 0
[2,5,4,1,7,3,6] => ?
=> ?
=> ? => ? = 0
[2,6,1,3,4,7,5] => ?
=> ?
=> ? => ? = 0
[2,6,1,5,3,7,4] => ?
=> ?
=> ? => ? = 0
[2,6,4,1,3,7,5] => ?
=> ?
=> ? => ? = 0
[2,6,4,5,1,7,3] => ?
=> ?
=> ? => ? = 0
[2,6,5,1,3,7,4] => ?
=> ?
=> ? => ? = 0
[2,6,7,1,3,4,5] => ?
=> ?
=> ? => ? = 0
[2,7,1,3,4,5,6] => ?
=> ?
=> ? => ? = 0
[2,7,1,3,6,4,5] => ?
=> ?
=> ? => ? = 0
[2,7,1,5,3,4,6] => ?
=> ?
=> ? => ? = 0
[2,7,1,5,6,3,4] => ?
=> ?
=> ? => ? = 0
[2,7,1,6,3,4,5] => ?
=> ?
=> ? => ? = 0
[2,7,4,1,3,5,6] => ?
=> ?
=> ? => ? = 0
[2,7,4,1,6,3,5] => ?
=> ?
=> ? => ? = 0
[2,7,4,5,1,3,6] => ?
=> ?
=> ? => ? = 0
[2,7,4,5,6,1,3] => ?
=> ?
=> ? => ? = 0
[2,7,4,6,1,3,5] => ?
=> ?
=> ? => ? = 0
[2,7,5,1,3,4,6] => ?
=> ?
=> ? => ? = 0
[2,7,5,1,6,3,4] => ?
=> ?
=> ? => ? = 0
[2,7,6,5,1,3,4] => ?
=> ?
=> ? => ? = 0
[3,1,4,5,6,7,2] => ?
=> ?
=> ? => ? = 0
[3,1,4,5,7,2,6] => ?
=> ?
=> ? => ? = 0
[3,1,4,6,2,7,5] => ?
=> ?
=> ? => ? = 0
[3,1,4,7,2,5,6] => ?
=> ?
=> ? => ? = 0
[3,1,4,7,6,2,5] => ?
=> ?
=> ? => ? = 0
[3,1,5,2,6,7,4] => ?
=> ?
=> ? => ? = 0
[3,1,5,2,7,4,6] => ?
=> ?
=> ? => ? = 0
[3,1,6,2,4,7,5] => ?
=> ?
=> ? => ? = 0
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000237
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00290: Parking functions —to ordered set partition⟶ Ordered set partitions
Mp00285: Ordered set partitions —to set partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00285: Ordered set partitions —to set partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [{1}] => {{1}}
=> [1] => 0
[1,1] => [{1,2}] => {{1,2}}
=> [2,1] => 1
[1,2] => [{1},{2}] => {{1},{2}}
=> [1,2] => 0
[2,1] => [{2},{1}] => {{1},{2}}
=> [1,2] => 0
[1,1,1] => [{1,2,3}] => {{1,2,3}}
=> [2,3,1] => 2
[1,1,2] => [{1,2},{3}] => {{1,2},{3}}
=> [2,1,3] => 1
[1,2,1] => [{1,3},{2}] => {{1,3},{2}}
=> [3,2,1] => 0
[2,1,1] => [{2,3},{1}] => {{1},{2,3}}
=> [1,3,2] => 1
[1,1,3] => [{1,2},{3}] => {{1,2},{3}}
=> [2,1,3] => 1
[1,3,1] => [{1,3},{2}] => {{1,3},{2}}
=> [3,2,1] => 0
[3,1,1] => [{2,3},{1}] => {{1},{2,3}}
=> [1,3,2] => 1
[1,2,2] => [{1},{2,3}] => {{1},{2,3}}
=> [1,3,2] => 1
[2,1,2] => [{2},{1,3}] => {{1,3},{2}}
=> [3,2,1] => 0
[2,2,1] => [{3},{1,2}] => {{1,2},{3}}
=> [2,1,3] => 1
[1,2,3] => [{1},{2},{3}] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => [{1},{3},{2}] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,1,3] => [{2},{1},{3}] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,3,1] => [{3},{1},{2}] => {{1},{2},{3}}
=> [1,2,3] => 0
[3,1,2] => [{2},{3},{1}] => {{1},{2},{3}}
=> [1,2,3] => 0
[3,2,1] => [{3},{2},{1}] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,1,1,1] => [{1,2,3,4}] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[1,1,1,2] => [{1,2,3},{4}] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[1,1,2,1] => [{1,2,4},{3}] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[1,2,1,1] => [{1,3,4},{2}] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[2,1,1,1] => [{2,3,4},{1}] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,1,1,3] => [{1,2,3},{4}] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[1,1,3,1] => [{1,2,4},{3}] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[1,3,1,1] => [{1,3,4},{2}] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[3,1,1,1] => [{2,3,4},{1}] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,1,1,4] => [{1,2,3},{4}] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[1,1,4,1] => [{1,2,4},{3}] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[1,4,1,1] => [{1,3,4},{2}] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[4,1,1,1] => [{2,3,4},{1}] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,1,2,2] => [{1,2},{3,4}] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[1,2,1,2] => [{1,3},{2,4}] => {{1,3},{2,4}}
=> [3,4,1,2] => 0
[1,2,2,1] => [{1,4},{2,3}] => {{1,4},{2,3}}
=> [4,3,2,1] => 1
[2,1,1,2] => [{2,3},{1,4}] => {{1,4},{2,3}}
=> [4,3,2,1] => 1
[2,1,2,1] => [{2,4},{1,3}] => {{1,3},{2,4}}
=> [3,4,1,2] => 0
[2,2,1,1] => [{3,4},{1,2}] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[1,1,2,3] => [{1,2},{3},{4}] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[1,1,3,2] => [{1,2},{4},{3}] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[1,2,1,3] => [{1,3},{2},{4}] => {{1,3},{2},{4}}
=> [3,2,1,4] => 0
[1,2,3,1] => [{1,4},{2},{3}] => {{1,4},{2},{3}}
=> [4,2,3,1] => 0
[1,3,1,2] => [{1,3},{4},{2}] => {{1,3},{2},{4}}
=> [3,2,1,4] => 0
[1,3,2,1] => [{1,4},{3},{2}] => {{1,4},{2},{3}}
=> [4,2,3,1] => 0
[2,1,1,3] => [{2,3},{1},{4}] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[2,1,3,1] => [{2,4},{1},{3}] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[2,3,1,1] => [{3,4},{1},{2}] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[3,1,1,2] => [{2,3},{4},{1}] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[3,1,2,1] => [{2,4},{3},{1}] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[2,3,4,5,6,7,1] => ? => ?
=> ? => ? = 0
[2,3,4,5,7,1,6] => ? => ?
=> ? => ? = 0
[2,3,4,6,1,7,5] => ? => ?
=> ? => ? = 0
[2,3,4,7,1,5,6] => ? => ?
=> ? => ? = 0
[2,3,4,7,6,1,5] => ? => ?
=> ? => ? = 0
[2,3,5,1,6,7,4] => ? => ?
=> ? => ? = 0
[2,3,5,1,7,4,6] => ? => ?
=> ? => ? = 0
[2,3,6,1,4,7,5] => ? => ?
=> ? => ? = 0
[2,3,6,5,1,7,4] => ? => ?
=> ? => ? = 0
[2,3,7,1,4,5,6] => ? => ?
=> ? => ? = 0
[2,3,7,1,6,4,5] => ? => ?
=> ? => ? = 0
[2,3,7,5,1,4,6] => ? => ?
=> ? => ? = 0
[2,3,7,5,6,1,4] => ? => ?
=> ? => ? = 0
[2,3,7,6,1,4,5] => ? => ?
=> ? => ? = 0
[2,4,1,5,6,7,3] => ? => ?
=> ? => ? = 0
[2,4,1,5,7,3,6] => ? => ?
=> ? => ? = 0
[2,4,1,6,3,7,5] => ? => ?
=> ? => ? = 0
[2,4,1,7,3,5,6] => ? => ?
=> ? => ? = 0
[2,4,1,7,6,3,5] => ? => ?
=> ? => ? = 0
[2,5,1,3,6,7,4] => ? => ?
=> ? => ? = 0
[2,5,1,3,7,4,6] => ? => ?
=> ? => ? = 0
[2,5,4,1,6,7,3] => ? => ?
=> ? => ? = 0
[2,5,4,1,7,3,6] => ? => ?
=> ? => ? = 0
[2,6,1,3,4,7,5] => ? => ?
=> ? => ? = 0
[2,6,1,5,3,7,4] => ? => ?
=> ? => ? = 0
[2,6,4,1,3,7,5] => ? => ?
=> ? => ? = 0
[2,6,4,5,1,7,3] => ? => ?
=> ? => ? = 0
[2,6,5,1,3,7,4] => ? => ?
=> ? => ? = 0
[2,6,7,1,3,4,5] => ? => ?
=> ? => ? = 0
[2,7,1,3,4,5,6] => ? => ?
=> ? => ? = 0
[2,7,1,3,6,4,5] => ? => ?
=> ? => ? = 0
[2,7,1,5,3,4,6] => ? => ?
=> ? => ? = 0
[2,7,1,5,6,3,4] => ? => ?
=> ? => ? = 0
[2,7,1,6,3,4,5] => ? => ?
=> ? => ? = 0
[2,7,4,1,3,5,6] => ? => ?
=> ? => ? = 0
[2,7,4,1,6,3,5] => ? => ?
=> ? => ? = 0
[2,7,4,5,1,3,6] => ? => ?
=> ? => ? = 0
[2,7,4,5,6,1,3] => ? => ?
=> ? => ? = 0
[2,7,4,6,1,3,5] => ? => ?
=> ? => ? = 0
[2,7,5,1,3,4,6] => ? => ?
=> ? => ? = 0
[2,7,5,1,6,3,4] => ? => ?
=> ? => ? = 0
[2,7,6,5,1,3,4] => ? => ?
=> ? => ? = 0
[3,1,4,5,6,7,2] => ? => ?
=> ? => ? = 0
[3,1,4,5,7,2,6] => ? => ?
=> ? => ? = 0
[3,1,4,6,2,7,5] => ? => ?
=> ? => ? = 0
[3,1,4,7,2,5,6] => ? => ?
=> ? => ? = 0
[3,1,4,7,6,2,5] => ? => ?
=> ? => ? = 0
[3,1,5,2,6,7,4] => ? => ?
=> ? => ? = 0
[3,1,5,2,7,4,6] => ? => ?
=> ? => ? = 0
[3,1,6,2,4,7,5] => ? => ?
=> ? => ? = 0
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000546
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [.,.]
=> [1] => 0
[1,1] => [[],[]]
=> [.,[.,.]]
=> [2,1] => 1
[1,2] => [[[]]]
=> [[.,.],.]
=> [1,2] => 0
[2,1] => [[[]]]
=> [[.,.],.]
=> [1,2] => 0
[1,1,1] => [[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 2
[1,1,2] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,2,1] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[2,1,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,1,3] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,3,1] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[3,1,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,2,2] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[2,2,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,2,3] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,3,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[2,1,3] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[2,3,1] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[3,1,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[3,2,1] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1,1,1] => [[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3
[1,1,1,2] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,2,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,2,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,1,3] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,3,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[3,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,1,4] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,4,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[4,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,2,2] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,2,1,2] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[1,2,2,1] => [[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[2,1,1,2] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,2,1] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[2,2,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,2,3] => [[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[1,1,3,2] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,2,1,3] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,3,1] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[1,3,1,2] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,3,2,1] => [[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[2,1,1,3] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,3,1] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[2,3,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[3,1,1,2] => [[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[3,1,2,1] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[2,3,4,5,6,7,1] => ?
=> ?
=> ? => ? = 0
[2,3,4,5,7,1,6] => ?
=> ?
=> ? => ? = 0
[2,3,4,6,1,7,5] => ?
=> ?
=> ? => ? = 0
[2,3,4,7,1,5,6] => ?
=> ?
=> ? => ? = 0
[2,3,4,7,6,1,5] => ?
=> ?
=> ? => ? = 0
[2,3,5,1,6,7,4] => ?
=> ?
=> ? => ? = 0
[2,3,5,1,7,4,6] => ?
=> ?
=> ? => ? = 0
[2,3,6,1,4,7,5] => ?
=> ?
=> ? => ? = 0
[2,3,6,5,1,7,4] => ?
=> ?
=> ? => ? = 0
[2,3,7,1,4,5,6] => ?
=> ?
=> ? => ? = 0
[2,3,7,1,6,4,5] => ?
=> ?
=> ? => ? = 0
[2,3,7,5,1,4,6] => ?
=> ?
=> ? => ? = 0
[2,3,7,5,6,1,4] => ?
=> ?
=> ? => ? = 0
[2,3,7,6,1,4,5] => ?
=> ?
=> ? => ? = 0
[2,4,1,5,6,7,3] => ?
=> ?
=> ? => ? = 0
[2,4,1,5,7,3,6] => ?
=> ?
=> ? => ? = 0
[2,4,1,6,3,7,5] => ?
=> ?
=> ? => ? = 0
[2,4,1,7,3,5,6] => ?
=> ?
=> ? => ? = 0
[2,4,1,7,6,3,5] => ?
=> ?
=> ? => ? = 0
[2,5,1,3,6,7,4] => ?
=> ?
=> ? => ? = 0
[2,5,1,3,7,4,6] => ?
=> ?
=> ? => ? = 0
[2,5,4,1,6,7,3] => ?
=> ?
=> ? => ? = 0
[2,5,4,1,7,3,6] => ?
=> ?
=> ? => ? = 0
[2,6,1,3,4,7,5] => ?
=> ?
=> ? => ? = 0
[2,6,1,5,3,7,4] => ?
=> ?
=> ? => ? = 0
[2,6,4,1,3,7,5] => ?
=> ?
=> ? => ? = 0
[2,6,4,5,1,7,3] => ?
=> ?
=> ? => ? = 0
[2,6,5,1,3,7,4] => ?
=> ?
=> ? => ? = 0
[2,6,7,1,3,4,5] => ?
=> ?
=> ? => ? = 0
[2,7,1,3,4,5,6] => ?
=> ?
=> ? => ? = 0
[2,7,1,3,6,4,5] => ?
=> ?
=> ? => ? = 0
[2,7,1,5,3,4,6] => ?
=> ?
=> ? => ? = 0
[2,7,1,5,6,3,4] => ?
=> ?
=> ? => ? = 0
[2,7,1,6,3,4,5] => ?
=> ?
=> ? => ? = 0
[2,7,4,1,3,5,6] => ?
=> ?
=> ? => ? = 0
[2,7,4,1,6,3,5] => ?
=> ?
=> ? => ? = 0
[2,7,4,5,1,3,6] => ?
=> ?
=> ? => ? = 0
[2,7,4,5,6,1,3] => ?
=> ?
=> ? => ? = 0
[2,7,4,6,1,3,5] => ?
=> ?
=> ? => ? = 0
[2,7,5,1,3,4,6] => ?
=> ?
=> ? => ? = 0
[2,7,5,1,6,3,4] => ?
=> ?
=> ? => ? = 0
[2,7,6,5,1,3,4] => ?
=> ?
=> ? => ? = 0
[3,1,4,5,6,7,2] => ?
=> ?
=> ? => ? = 0
[3,1,4,5,7,2,6] => ?
=> ?
=> ? => ? = 0
[3,1,4,6,2,7,5] => ?
=> ?
=> ? => ? = 0
[3,1,4,7,2,5,6] => ?
=> ?
=> ? => ? = 0
[3,1,4,7,6,2,5] => ?
=> ?
=> ? => ? = 0
[3,1,5,2,6,7,4] => ?
=> ?
=> ? => ? = 0
[3,1,5,2,7,4,6] => ?
=> ?
=> ? => ? = 0
[3,1,6,2,4,7,5] => ?
=> ?
=> ? => ? = 0
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St001640
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [.,.]
=> [1] => 0
[1,1] => [[],[]]
=> [[.,.],.]
=> [1,2] => 1
[1,2] => [[[]]]
=> [.,[.,.]]
=> [2,1] => 0
[2,1] => [[[]]]
=> [.,[.,.]]
=> [2,1] => 0
[1,1,1] => [[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 2
[1,1,2] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,2,1] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[2,1,1] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,3] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,3,1] => [[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[3,1,1] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,2,2] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[2,1,2] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[2,2,1] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,2,3] => [[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[1,3,2] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[2,1,3] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[2,3,1] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[3,1,2] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[3,2,1] => [[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[1,1,1,1] => [[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 3
[1,1,1,2] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,2,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,2,1,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[2,1,1,1] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,1,3] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,3,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,3,1,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[3,1,1,1] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,1,4] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,4,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,4,1,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[4,1,1,1] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,2,2] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,2,1,2] => [[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[1,2,2,1] => [[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[2,1,1,2] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[2,1,2,1] => [[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[2,2,1,1] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,2,3] => [[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[1,1,3,2] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,2,1,3] => [[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,2,3,1] => [[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[1,3,1,2] => [[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,3,2,1] => [[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 0
[2,1,1,3] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[2,1,3,1] => [[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[2,3,1,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[3,1,1,2] => [[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[3,1,2,1] => [[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[2,3,4,5,6,7,1] => ?
=> ?
=> ? => ? = 0
[2,3,4,5,7,1,6] => ?
=> ?
=> ? => ? = 0
[2,3,4,6,1,7,5] => ?
=> ?
=> ? => ? = 0
[2,3,4,7,1,5,6] => ?
=> ?
=> ? => ? = 0
[2,3,4,7,6,1,5] => ?
=> ?
=> ? => ? = 0
[2,3,5,1,6,7,4] => ?
=> ?
=> ? => ? = 0
[2,3,5,1,7,4,6] => ?
=> ?
=> ? => ? = 0
[2,3,6,1,4,7,5] => ?
=> ?
=> ? => ? = 0
[2,3,6,5,1,7,4] => ?
=> ?
=> ? => ? = 0
[2,3,7,1,4,5,6] => ?
=> ?
=> ? => ? = 0
[2,3,7,1,6,4,5] => ?
=> ?
=> ? => ? = 0
[2,3,7,5,1,4,6] => ?
=> ?
=> ? => ? = 0
[2,3,7,5,6,1,4] => ?
=> ?
=> ? => ? = 0
[2,3,7,6,1,4,5] => ?
=> ?
=> ? => ? = 0
[2,4,1,5,6,7,3] => ?
=> ?
=> ? => ? = 0
[2,4,1,5,7,3,6] => ?
=> ?
=> ? => ? = 0
[2,4,1,6,3,7,5] => ?
=> ?
=> ? => ? = 0
[2,4,1,7,3,5,6] => ?
=> ?
=> ? => ? = 0
[2,4,1,7,6,3,5] => ?
=> ?
=> ? => ? = 0
[2,5,1,3,6,7,4] => ?
=> ?
=> ? => ? = 0
[2,5,1,3,7,4,6] => ?
=> ?
=> ? => ? = 0
[2,5,4,1,6,7,3] => ?
=> ?
=> ? => ? = 0
[2,5,4,1,7,3,6] => ?
=> ?
=> ? => ? = 0
[2,6,1,3,4,7,5] => ?
=> ?
=> ? => ? = 0
[2,6,1,5,3,7,4] => ?
=> ?
=> ? => ? = 0
[2,6,4,1,3,7,5] => ?
=> ?
=> ? => ? = 0
[2,6,4,5,1,7,3] => ?
=> ?
=> ? => ? = 0
[2,6,5,1,3,7,4] => ?
=> ?
=> ? => ? = 0
[2,6,7,1,3,4,5] => ?
=> ?
=> ? => ? = 0
[2,7,1,3,4,5,6] => ?
=> ?
=> ? => ? = 0
[2,7,1,3,6,4,5] => ?
=> ?
=> ? => ? = 0
[2,7,1,5,3,4,6] => ?
=> ?
=> ? => ? = 0
[2,7,1,5,6,3,4] => ?
=> ?
=> ? => ? = 0
[2,7,1,6,3,4,5] => ?
=> ?
=> ? => ? = 0
[2,7,4,1,3,5,6] => ?
=> ?
=> ? => ? = 0
[2,7,4,1,6,3,5] => ?
=> ?
=> ? => ? = 0
[2,7,4,5,1,3,6] => ?
=> ?
=> ? => ? = 0
[2,7,4,5,6,1,3] => ?
=> ?
=> ? => ? = 0
[2,7,4,6,1,3,5] => ?
=> ?
=> ? => ? = 0
[2,7,5,1,3,4,6] => ?
=> ?
=> ? => ? = 0
[2,7,5,1,6,3,4] => ?
=> ?
=> ? => ? = 0
[2,7,6,5,1,3,4] => ?
=> ?
=> ? => ? = 0
[3,1,4,5,6,7,2] => ?
=> ?
=> ? => ? = 0
[3,1,4,5,7,2,6] => ?
=> ?
=> ? => ? = 0
[3,1,4,6,2,7,5] => ?
=> ?
=> ? => ? = 0
[3,1,4,7,2,5,6] => ?
=> ?
=> ? => ? = 0
[3,1,4,7,6,2,5] => ?
=> ?
=> ? => ? = 0
[3,1,5,2,6,7,4] => ?
=> ?
=> ? => ? = 0
[3,1,5,2,7,4,6] => ?
=> ?
=> ? => ? = 0
[3,1,6,2,4,7,5] => ?
=> ?
=> ? => ? = 0
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [.,.]
=> [1] => 1 = 0 + 1
[1,1] => [[],[]]
=> [.,[.,.]]
=> [2,1] => 2 = 1 + 1
[1,2] => [[[]]]
=> [[.,.],.]
=> [1,2] => 1 = 0 + 1
[2,1] => [[[]]]
=> [[.,.],.]
=> [1,2] => 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[2,3,4,5,6,7,1] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,4,5,7,1,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,4,6,1,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,4,7,1,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,4,7,6,1,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,5,1,6,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,5,1,7,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,6,1,4,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,6,5,1,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,1,4,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,1,6,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,5,1,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,5,6,1,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,6,1,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,5,6,7,3] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,5,7,3,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,6,3,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,7,3,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,7,6,3,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,5,1,3,6,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,5,1,3,7,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,5,4,1,6,7,3] => ?
=> ?
=> ? => ? = 0 + 1
[2,5,4,1,7,3,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,1,3,4,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,1,5,3,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,4,1,3,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,4,5,1,7,3] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,5,1,3,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,7,1,3,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,3,4,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,3,6,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,5,3,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,5,6,3,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,6,3,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,1,3,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,1,6,3,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,5,1,3,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,5,6,1,3] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,6,1,3,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,5,1,3,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,5,1,6,3,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,6,5,1,3,4] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,5,6,7,2] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,5,7,2,6] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,6,2,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,7,2,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,7,6,2,5] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,5,2,6,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,5,2,7,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,6,2,4,7,5] => ?
=> ?
=> ? => ? = 0 + 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000025
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[1,2] => [[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,4,5,6,7,1] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,4,5,7,1,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,4,6,1,7,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,4,7,1,5,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,4,7,6,1,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,5,1,6,7,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,5,1,7,4,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,6,1,4,7,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,6,5,1,7,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,7,1,4,5,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,7,1,6,4,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,7,5,1,4,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,7,5,6,1,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,3,7,6,1,4,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,4,1,5,6,7,3] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,4,1,5,7,3,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,4,1,6,3,7,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,4,1,7,3,5,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,4,1,7,6,3,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,5,1,3,6,7,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,5,1,3,7,4,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,5,4,1,6,7,3] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,5,4,1,7,3,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,6,1,3,4,7,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,6,1,5,3,7,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,6,4,1,3,7,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,6,4,5,1,7,3] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,6,5,1,3,7,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,6,7,1,3,4,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,1,3,4,5,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,1,3,6,4,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,1,5,3,4,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,1,5,6,3,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,1,6,3,4,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,4,1,3,5,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,4,1,6,3,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,4,5,1,3,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,4,5,6,1,3] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,4,6,1,3,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,5,1,3,4,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,5,1,6,3,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[2,7,6,5,1,3,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[3,1,4,5,6,7,2] => ?
=> ?
=> ?
=> ? = 0 + 1
[3,1,4,5,7,2,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[3,1,4,6,2,7,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[3,1,4,7,2,5,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[3,1,4,7,6,2,5] => ?
=> ?
=> ?
=> ? = 0 + 1
[3,1,5,2,6,7,4] => ?
=> ?
=> ?
=> ? = 0 + 1
[3,1,5,2,7,4,6] => ?
=> ?
=> ?
=> ? = 0 + 1
[3,1,6,2,4,7,5] => ?
=> ?
=> ?
=> ? = 0 + 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000056
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [1,0]
=> [1] => 1 = 0 + 1
[1,1] => [[],[]]
=> [1,0,1,0]
=> [1,2] => 2 = 1 + 1
[1,2] => [[[]]]
=> [1,1,0,0]
=> [2,1] => 1 = 0 + 1
[2,1] => [[[]]]
=> [1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 1
[2,3,4,5,6,7,1] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,4,5,7,1,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,4,6,1,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,4,7,1,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,4,7,6,1,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,5,1,6,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,5,1,7,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,6,1,4,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,6,5,1,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,1,4,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,1,6,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,5,1,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,5,6,1,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,3,7,6,1,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,5,6,7,3] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,5,7,3,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,6,3,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,7,3,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,4,1,7,6,3,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,5,1,3,6,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,5,1,3,7,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,5,4,1,6,7,3] => ?
=> ?
=> ? => ? = 0 + 1
[2,5,4,1,7,3,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,1,3,4,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,1,5,3,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,4,1,3,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,4,5,1,7,3] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,5,1,3,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,6,7,1,3,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,3,4,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,3,6,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,5,3,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,5,6,3,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,1,6,3,4,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,1,3,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,1,6,3,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,5,1,3,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,5,6,1,3] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,4,6,1,3,5] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,5,1,3,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,5,1,6,3,4] => ?
=> ?
=> ? => ? = 0 + 1
[2,7,6,5,1,3,4] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,5,6,7,2] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,5,7,2,6] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,6,2,7,5] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,7,2,5,6] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,4,7,6,2,5] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,5,2,6,7,4] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,5,2,7,4,6] => ?
=> ?
=> ? => ? = 0 + 1
[3,1,6,2,4,7,5] => ?
=> ?
=> ? => ? = 0 + 1
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000843The decomposition number of a perfect matching. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St000439The position of the first down step of a Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000502The number of successions of a set partitions. St000061The number of nodes on the left branch of a binary tree. St000989The number of final rises of a permutation. St000504The cardinality of the first block of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
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