searching the database
Your data matches 722 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001020
(load all 1222 compositions to match this statistic)
(load all 1222 compositions to match this statistic)
St001020: 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]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> 5
[1,1,1,0,1,1,0,0,0,0]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> 5
[1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> 6
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000395
(load all 565 compositions to match this statistic)
(load all 565 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000395: 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]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000998
(load all 569 compositions to match this statistic)
(load all 569 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000998: 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]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7 = 6 + 1
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
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: St000026
(load all 604 compositions to match this statistic)
(load all 604 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
Description
The position of the first return of a Dyck path.
Matching statistic: St000145
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2]
=> 1
[1,0,1,0]
=> [3,1,2] => [3]
=> 2
[1,1,0,0]
=> [2,3,1] => [3]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> 3
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> 4
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> 4
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> 4
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> 4
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> 4
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> 4
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> 4
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> 5
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [6]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [6]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [6]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6]
=> 5
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6]
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [7]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7]
=> 6
Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000228
(load all 895 compositions to match this statistic)
(load all 895 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> 2
[1,1,0,0]
=> [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 3
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 3
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 3
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 4
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 4
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 4
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 4
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 4
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 4
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [2,2,1]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 5
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 5
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [2,2,1,1]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [2,2,1,1]
=> 6
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2]
=> 1
[1,0,1,0]
=> [3,1,2] => [3]
=> 2
[1,1,0,0]
=> [2,3,1] => [3]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> 3
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> 4
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> 4
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> 4
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> 4
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> 4
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> 4
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> 4
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> 5
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [6]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [6]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [6]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6]
=> 5
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6]
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [7]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7]
=> 6
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2]
=> 1
[1,0,1,0]
=> [3,1,2] => [3]
=> 2
[1,1,0,0]
=> [2,3,1] => [3]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> 3
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> 4
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> 4
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> 4
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> 4
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> 4
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> 4
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> 4
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> 5
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [6]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [6]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [6]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6]
=> 5
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6]
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [7]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7]
=> 6
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000394
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: 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
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 2
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
The following 712 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000445The number of rises of length 1 of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000676The number of odd rises of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000728The dimension of a set partition. St000839The largest opener of a set partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001034The area of the parallelogram polyomino associated with the Dyck path. 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. 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:
St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001523The degree of symmetry of a Dyck path. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000053The number of valleys of the Dyck path. St000081The number of edges of a graph. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000531The leading coefficient of the rook polynomial of an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000784The maximum of the length and the largest part of the integer partition. St000823The number of unsplittable factors of the set partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000971The smallest closer of a set partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001062The maximal size of a block of a set partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001075The minimal size of a block of a set partition. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001479The number of bridges of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001955The number of natural descents for set-valued two row standard Young tableaux. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001400The total number of Littlewood-Richardson tableaux of given shape. St001814The number of partitions interlacing the given partition. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000022The number of fixed points of a permutation. St000025The number of initial rises of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000054The first entry of the permutation. St000058The order of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000141The maximum drop size of a permutation. St000153The number of adjacent cycles of a permutation. St000167The number of leaves of an ordered tree. St000203The number of external nodes of a binary tree. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000258The burning number of a graph. St000273The domination number of a graph. St000293The number of inversions of a binary word. St000308The height of the tree associated to a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000400The path length of an ordered tree. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000482The (zero)-forcing number of a graph. St000543The size of the conjugacy class of a binary word. St000544The cop number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000617The number of global maxima of a Dyck path. St000626The minimal period of a binary word. St000655The length of the minimal rise of a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000703The number of deficiencies of a permutation. St000721The sum of the partition sizes in the oscillating tableau corresponding to a perfect matching. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744The length of the path to the largest entry in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000759The smallest missing part in an integer partition. St000765The number of weak records in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000863The length of the first row of the shifted shape of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000883The number of longest increasing subsequences of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000921The number of internal inversions of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001093The detour number of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001463The number of distinct columns in the nullspace of a graph. St001497The position of the largest weak excedence of a permutation. St001554The number of distinct nonempty subtrees of a binary tree. St001566The length of the longest arithmetic progression in a permutation. St001616The number of neutral elements in a lattice. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001720The minimal length of a chain of small intervals in a lattice. St001746The coalition number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001820The size of the image of the pop stack sorting operator. St001829The common independence number of a graph. St001910The height of the middle non-run of a Dyck path. St000010The length of the partition. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000259The diameter of a connected graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000359The number of occurrences of the pattern 23-1. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000410The tree factorial of an ordered tree. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000475The number of parts equal to 1 in a partition. St000507The number of ascents of a standard tableau. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000546The number of global descents of a permutation. St000657The smallest part of an integer composition. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000731The number of double exceedences of a permutation. St000734The last entry in the first row of a standard tableau. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000808The number of up steps of the associated bargraph. St000825The sum of the major and the inverse major index of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000922The minimal number such that all substrings of this length are unique. St000979Half of MacMahon's equal index of a Dyck path. St000982The length of the longest constant subword. St000993The multiplicity of the largest part of an integer partition. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001096The size of the overlap set of a permutation. St001127The sum of the squares of the parts of a partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001298The number of repeated entries in the Lehmer code of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001405The number of bonds in a permutation. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001462The number of factors of a standard tableaux under concatenation. St001484The number of singletons of an integer partition. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001622The number of join-irreducible elements of a lattice. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001759The Rajchgot index of a permutation. St001777The number of weak descents in an integer composition. St001933The largest multiplicity of a part in an integer partition. St001949The rigidity index of a graph. St000393The number of strictly increasing runs in a binary word. St000806The semiperimeter of the associated bargraph. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000247The number of singleton blocks of a set partition. St000294The number of distinct factors of a binary word. St000438The position of the last up step in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000518The number of distinct subsequences in a binary word. St000625The sum of the minimal distances to a greater element. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000729The minimal arc length of a set partition. St000873The aix statistic of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000925The number of topologically connected components of a set partition. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001074The number of inversions of the cyclic embedding of a permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000189The number of elements in the poset. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000296The length of the symmetric border of a binary word. St000385The number of vertices with out-degree 1 in a binary tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000442The maximal area to the right of an up step of a Dyck path. St000520The number of patterns in a permutation. St000627The exponent of a binary word. St000874The position of the last double rise in a Dyck path. St000984The number of boxes below precisely one peak. St000989The number of final rises of a permutation. St001052The length of the exterior of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001430The number of positive entries in a signed permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000837The number of ascents of distance 2 of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001651The Frankl number of a lattice. St001958The degree of the polynomial interpolating the values of a permutation. St001875The number of simple modules with projective dimension at most 1. St001246The maximal difference between two consecutive entries of a permutation. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000738The first entry in the last row of a standard tableau. St001925The minimal number of zeros in a row of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St000651The maximal size of a rise in a permutation. St000653The last descent of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000050The depth or height of a binary tree. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000225Difference between largest and smallest parts in a partition. St000144The pyramid weight of the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000015The number of peaks of a Dyck path. St000211The rank of the set partition. St000501The size of the first part in the decomposition of a permutation. St000553The number of blocks of a graph. St000733The row containing the largest entry of a standard tableau. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules 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. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000051The size of the left subtree of a binary tree. St000209Maximum difference of elements in cycles. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. 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. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000056The decomposition (or block) number of a permutation. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000062The length of the longest increasing subsequence of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000084The number of subtrees. St000117The number of centered tunnels of a Dyck path. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000171The degree of the graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000221The number of strong fixed points of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000241The number of cyclical small excedances. St000291The number of descents of a binary word. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000338The number of pixed points of a permutation. St000363The number of minimal vertex covers of a graph. St000389The number of runs of ones of odd length in a binary word. 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. St000542The number of left-to-right-minima of a permutation. St000632The jump number of the poset. St000638The number of up-down runs of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000730The maximal arc length of a set partition. St000746The number of pairs with odd minimum in a perfect matching. St000843The decomposition number of a perfect matching. St000895The number of ones on the main diagonal of an alternating sign matrix. St000924The number of topologically connected components of a perfect matching. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001120The length of a longest path in a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001252Half the sum of the even parts of a partition. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000004The major index of a permutation. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000067The inversion number of the alternating sign matrix. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000210Minimum over maximum difference of elements in cycles. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000292The number of ascents of a binary word. St000312The number of leaves in a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000362The size of a minimal vertex cover of a graph. St000383The last part of an integer composition. St000446The disorder of a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000527The width of the poset. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000840The number of closers smaller than the largest opener in a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St001065Number of indecomposable reflexive modules 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. St001172The number of 1-rises at odd height of a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001391The disjunction number of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001614The cyclic permutation representation number of a skew partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001726The number of visible inversions of a permutation. St001781The interlacing number of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001658The total number of rook placements on a Ferrers board. St001480The number of simple summands of the module J^2/J^3. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000061The number of nodes on the left branch of a binary tree. St000064The number of one-box pattern of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000673The number of non-fixed points of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000894The trace of an alternating sign matrix. St000990The first ascent of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001959The product of the heights of the peaks of a Dyck path. St000060The greater neighbor of the maximum. St000083The number of left oriented leafs of a binary tree except the first one. St000197The number of entries equal to positive one in the alternating sign matrix. St000216The absolute length of a permutation. St000354The number of recoils of a permutation. St000530The number of permutations with the same descent word as the given permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000836The number of descents of distance 2 of a permutation. St001280The number of parts of an integer partition that are at least two. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001725The harmonious chromatic number of a graph. St000080The rank of the poset. St001917The order of toric promotion on the set of labellings of a graph. St001268The size of the largest ordinal summand in the poset. St001645The pebbling number of a connected graph. St000287The number of connected components of a graph. St001828The Euler characteristic of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St000656The number of cuts of a poset. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St000246The number of non-inversions of a permutation. St001644The dimension of a graph. St001468The smallest fixpoint of a permutation. 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. St000231Sum of the maximal elements of the blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000134The size of the orbit of an alternating sign matrix under gyration. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000305The inverse major index of a permutation. St000332The positive inversions of an alternating sign matrix. St000450The number of edges minus the number of vertices plus 2 of a graph. St000547The number of even non-empty partial sums of an integer partition. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has 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. St001304The number of maximally independent sets of vertices of a graph. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001429The number of negative entries in a signed permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001671Haglund's hag of a permutation. St001963The tree-depth of a graph. St000039The number of crossings of a permutation. St000095The number of triangles of a graph. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000159The number of distinct parts of the integer partition. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000369The dinv deficit of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000549The number of odd partial sums of an integer partition. St000732The number of double deficiencies of a permutation. St000947The major index east count of a Dyck path. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001161The major index north count of a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001379The number of inversions plus the major index of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001684The reduced word complexity of a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000456The monochromatic index of a connected graph. St000529The number of permutations whose descent word is the given binary word. St000946The sum of the skew hook positions in a Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001808The box weight or horizontal decoration of a Dyck path. St001706The number of closed sets in a graph. St000327The number of cover relations in a poset. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001432The order dimension of the partition. St001948The number of augmented double ascents of a permutation. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. 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 $eA$. St001927Sparre Andersen's number of positives of a signed permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000822The Hadwiger number of the graph. St001316The domatic number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000741The Colin de Verdière graph invariant. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000691The number of changes of a binary word. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001845The number of join irreducibles minus the rank of a lattice. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001812The biclique partition number of a graph. St000352The Elizalde-Pak rank of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001861The number of Bruhat lower covers of a permutation. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001782The order of rowmotion on the set of order ideals of a poset. St000044The number of vertices of the unicellular map given by a perfect matching. St000135The number of lucky cars of the parking function. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000017The number of inversions of a standard tableau. St000820The number of compositions obtained by rotating the composition. St000035The number of left outer peaks of a permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001214The aft of an integer partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001413Half the length of the longest even length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001427The number of descents of a signed permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001524The degree of symmetry of a binary word. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000834The number of right outer peaks of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001555The order of a signed permutation. St001712The number of natural descents of a standard Young tableau. St001742The difference of the maximal and the minimal degree in a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000014The number of parking functions supported by a Dyck path. St000242The number of indices that are not cyclical small weak excedances. St001486The number of corners of the ribbon associated with an integer composition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001926Sparre Andersen's position of the maximum of a signed permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!