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Your data matches 244 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St001184
(load all 164 compositions to match this statistic)
(load all 164 compositions to match this statistic)
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 1
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 4
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St000007
(load all 191 compositions to match this statistic)
(load all 191 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding 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
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 1
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 4
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: St000011
(load all 100 compositions to match this statistic)
(load all 100 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
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: St000025
(load all 54 compositions to match this statistic)
(load all 54 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ 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
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
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: St000068
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> ([],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> ([],3)
=> 3
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 4
Description
The number of minimal elements in a poset.
Matching statistic: St000314
(load all 92 compositions to match this statistic)
(load all 92 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 2
[1,0,1,0,1,0]
=> [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 4
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000838
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000838: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000838: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [(1,2)]
=> 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 2
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 3
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 2
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 3
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 3
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 4
Description
The number of terminal right-hand endpoints when the vertices are written in order.
An opener (or left hand endpoint) of a perfect matching is a number that is matched with a larger number, which is then called a closer (or right hand endpoint).
The opener-closer sequence of the perfect matching $\{(1,3),(2,5),(4,6)\}$ is $OOCOCC$, so the number of terminal right-hand endpoints is $2$.
The number of perfect matchings of $\{1,\dots,2n\}$ with exactly $T$ terminal closers, according to [1] computed in [2], is
$$
\frac{T(2n-T-1)!}{2^{n-T}(n-T)!}.
$$
Matching statistic: St000439
(load all 54 compositions to match this statistic)
(load all 54 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ 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
[1,0]
=> [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St001185
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
Matching statistic: St000015
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 4
Description
The number of peaks of a Dyck path.
The following 234 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000759The smallest missing part in an integer partition. St000843The decomposition number of a perfect matching. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000053The number of valleys of the Dyck path. St000133The "bounce" of a permutation. St000203The number of external nodes of a binary tree. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000331The number of upper interactions of a Dyck path. St000504The cardinality of the first block of a set partition. St000546The number of global descents of a permutation. St000738The first entry in the last row of a standard tableau. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000010The length of the partition. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000239The number of small weak excedances. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000502The number of successions of a set partitions. St000505The biggest entry in the block containing the 1. St000544The cop number of a graph. St000700The protection number of an ordered tree. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000822The Hadwiger number of the graph. St000883The number of longest increasing subsequences of a permutation. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000916The packing number of a graph. St000996The number of exclusive left-to-right maxima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001829The common independence number of a graph. St001963The tree-depth of a graph. St000005The bounce statistic of a Dyck path. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000245The number of ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000445The number of rises of length 1 of a Dyck path. St000446The disorder of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000675The number of centered multitunnels of a Dyck path. St000734The last entry in the first row of a standard tableau. St000783The side length of the largest staircase partition fitting into a partition. St000823The number of unsplittable factors of the set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001358The largest degree of a regular subgraph of a graph. St001484The number of singletons of an integer partition. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001777The number of weak descents in an integer composition. St000654The first descent of a permutation. St000989The number of final rises of a permutation. St000061The number of nodes on the left branch of a binary tree. St000717The number of ordinal summands of a poset. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000420The number of Dyck paths that are weakly above a Dyck path. St000906The length of the shortest maximal chain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000653The last descent of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St001480The number of simple summands of the module J^2/J^3. St000159The number of distinct parts of the integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000703The number of deficiencies of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001432The order dimension of the partition. St000648The number of 2-excedences of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000924The number of topologically connected components of a perfect matching. 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. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. 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. St000181The number of connected components of the Hasse diagram for the poset. St000702The number of weak deficiencies of a permutation. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000338The number of pixed points of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000756The sum of the positions of the left to right maxima of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000898The number of maximal entries in the last diagonal of the monotone triangle. St001889The size of the connectivity set of a signed permutation. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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. St001904The length of the initial strictly increasing segment of a parking function. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001937The size of the center of a parking function. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001712The number of natural descents of a standard Young tableau. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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