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Your data matches 155 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,1] => [1,2] => 1 = 0 + 1
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => 2 = 1 + 1
[1,1,1] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [1,2,3] => 1 = 0 + 1
[1,2,1] => [1,2,3] => 1 = 0 + 1
[2,1,1] => [2,1,3] => 1 = 0 + 1
[1,1,3] => [1,2,3] => 1 = 0 + 1
[1,3,1] => [1,3,2] => 2 = 1 + 1
[3,1,1] => [2,3,1] => 2 = 1 + 1
[1,2,2] => [1,2,3] => 1 = 0 + 1
[2,1,2] => [2,1,3] => 1 = 0 + 1
[2,2,1] => [3,1,2] => 2 = 1 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 1 = 0 + 1
[2,3,1] => [3,1,2] => 2 = 1 + 1
[3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1] => [3,2,1] => 3 = 2 + 1
[1,1,1,1] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,2] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,1] => [1,2,3,4] => 1 = 0 + 1
[2,1,1,1] => [2,1,3,4] => 1 = 0 + 1
[1,1,1,3] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,1] => [1,2,3,4] => 1 = 0 + 1
[1,3,1,1] => [1,3,2,4] => 1 = 0 + 1
[3,1,1,1] => [2,3,1,4] => 1 = 0 + 1
[1,1,1,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,2] => [1,2,3,4] => 1 = 0 + 1
[1,2,2,1] => [1,2,3,4] => 1 = 0 + 1
[2,1,1,2] => [2,1,3,4] => 1 = 0 + 1
[2,1,2,1] => [2,1,3,4] => 1 = 0 + 1
[2,2,1,1] => [3,1,2,4] => 1 = 0 + 1
[1,1,2,3] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,2] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,3] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,1] => [1,2,3,4] => 1 = 0 + 1
[1,3,1,2] => [1,3,2,4] => 1 = 0 + 1
[1,3,2,1] => [1,3,2,4] => 1 = 0 + 1
[2,1,1,3] => [2,1,3,4] => 1 = 0 + 1
[2,1,3,1] => [2,1,3,4] => 1 = 0 + 1
[2,3,1,1] => [3,1,2,4] => 1 = 0 + 1
[3,1,1,2] => [2,3,1,4] => 1 = 0 + 1
[3,1,2,1] => [2,3,1,4] => 1 = 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 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 0 + 1
[1,1] => [1,1,0,0]
=> 2 = 1 + 1
[1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,0,1,0]
=> 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 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: St000084
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
Description
The number of subtrees.
Matching statistic: St000439
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 2 = 0 + 2
[1,1] => [1,1,0,0]
=> 3 = 1 + 2
[1,2] => [1,0,1,0]
=> 2 = 0 + 2
[2,1] => [1,0,1,0]
=> 2 = 0 + 2
[1,1,1] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[1,1,2] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,2,1] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,1] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,3] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[1,3,1] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,1] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[1,2,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,2,1] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[2,1,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[2,3,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[3,1,2] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[3,2,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St001640
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,1,0,0]
=> [1,2] => 1
[1,2] => [1,0,1,0]
=> [2,1] => 0
[2,1] => [1,0,1,0]
=> [2,1] => 0
[1,1,1] => [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,1,2] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,3,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[3,1,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,2,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,2,1] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,3,2] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,1,3] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,3,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,1,2] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,2,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000011
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 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: St000054
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1 = 0 + 1
[1,1] => [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[1,2] => [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[2,1] => [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> [3,2,1] => 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> [1,3,2] => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000066
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [[1]]
=> 1 = 0 + 1
[1,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2 = 1 + 1
[1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1 = 0 + 1
[2,1] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
Matching statistic: St000069
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => ([],2)
=> 2 = 1 + 1
[1,1,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[3,1,1] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,2,1] => [3,1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,3,1] => [3,1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [3,2,1] => ([],3)
=> 3 = 2 + 1
[1,1,1,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,1,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,1,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,1,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,1,1,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,4,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1 = 0 + 1
[2,1,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1 = 0 + 1
[2,2,1,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,3,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,1,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,1,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,1,3] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[3,1,1,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[3,1,2,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
Description
The number of maximal elements of a poset.
Matching statistic: St000297
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00093: Dyck paths —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 10 => 1 = 0 + 1
[1,1] => [1,1,0,0]
=> 1100 => 2 = 1 + 1
[1,2] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[2,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> 111000 => 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> 110010 => 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> 110010 => 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> 110010 => 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> 11101000 => 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> 11101000 => 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> 11101000 => 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> 11101000 => 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> 11100010 => 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> 11011000 => 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> 11011000 => 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> 11011000 => 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> 11011000 => 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> 11011000 => 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> 11011000 => 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
Description
The number of leading ones in a binary word.
The following 145 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000971The smallest closer of a set partition. 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. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000738The first entry in the last row of a standard tableau. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000133The "bounce" of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000546The number of global descents of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000068The number of minimal elements in a poset. St000213The number of weak exceedances (also weak excedences) of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000352The Elizalde-Pak rank of a permutation. St000383The last part of an integer composition. St000505The biggest entry in the block containing the 1. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000759The smallest missing part in an integer partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000203The number of external nodes of a binary tree. St000734The last entry in the first row of a standard tableau. 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. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St000061The number of nodes on the left branch of a binary tree. St000504The cardinality of the first block of a set partition. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000717The number of ordinal summands of a poset. St000823The number of unsplittable factors of the set partition. St000990The first ascent of a permutation. St001525The number of symmetric hooks on the diagonal of a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000137The Grundy value of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000934The 2-degree of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000929The constant term of the character polynomial of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001651The Frankl number of a lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000115The single entry in the last row. St000259The diameter of a connected graph. St001947The number of ties in a parking function. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000455The second largest eigenvalue of a graph if it is integral. St001937The size of the center of a parking function. St000441The number of successions of a permutation. St001330The hat guessing number of a graph. 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. St000160The multiplicity of the smallest part of a partition. St000475The number of parts equal to 1 in a partition. St000177The number of free tiles in the pattern. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000445The number of rises of length 1 of a Dyck path. St000904The maximal number of repetitions of an integer composition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000674The number of hills of a Dyck path. St000942The number of critical left to right maxima of the parking functions. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001487The number of inner corners of a skew partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000090The variation of a composition. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000839The largest opener of a set partition. St000230Sum of the minimal elements of the blocks of a set partition.
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