Your data matches 47 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001182
(load all 14 compositions to match this statistic)
Mapping
St001182: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => 3 = 2 + 1
[1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
Description
Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000235
(load all 12 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => 2
[1,1,0,0]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => 3
[1,0,1,1,0,0]
 => [1,3,2] => 2
[1,1,0,0,1,0]
 => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => 0
[1,1,1,0,0,0]
 => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3
[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] => 4
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[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] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 4
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 5
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 3
[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] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 5
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 5
[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] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 3
[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] => 4
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 3
[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] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 4
[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]
 => [2,5,3,4,1] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 5
Description
The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
Matching statistic: St000673
(load all 27 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => 0
[1,1,0,0]
 => [2,1] => 2
[1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => 2
[1,1,0,0,1,0]
 => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => 3
[1,1,1,0,0,0]
 => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[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] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 4
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 4
[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] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 2
[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] => 0
[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] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 4
[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] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 2
[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] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 5
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 5
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 5
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 2
Description
The number of non-fixed points of a permutation. In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St001005
(load all 6 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [2,1] => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => 3
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 4
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,2,3,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,2,4,1] => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,3,1] => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,3,4,1] => 4
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,4,1,2] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,3,5,2] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,4,2] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,3,4,5,2] => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,5,2,3] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => 5
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => 4
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,2,3,5,1] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,2,4,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,2,5,4,1] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,2,4,5,1] => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,2,5,1,3] => 4
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,3,1,4,5] => 3
Description
The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.
Matching statistic: St001279
(load all 2 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [1,1]
 => 0
[1,1,0,0]
 => [2,1] => [2]
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3]
 => 3
[1,1,1,0,0,0]
 => [3,2,1] => [2,1]
 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 4
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 4
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,1,1,1]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,1,1]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,1,1,1]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,2,1]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,1]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,1]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,2]
 => 5
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,1,1,1]
 => 2
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001458
(load all 11 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001458: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 2
[1,1,0,0]
 => [1,2] => ([],2)
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
Description
The rank of the adjacency matrix of a graph.
Matching statistic: St001255
(load all 5 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1] => [1,0,1,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [2] => [1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
Description
The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St001504
(load all 7 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2 = 0 + 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5 = 3 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 2 + 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 5 = 3 + 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 2 + 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 4 + 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 2 + 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6 = 4 + 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5 = 3 + 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 5 = 3 + 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6 = 4 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5 = 3 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 6 = 4 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 6 = 4 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 5 = 3 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 5 = 3 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 6 = 4 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4 = 2 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 6 = 4 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 6 = 4 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 6 = 4 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 7 = 5 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 6 = 4 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 6 = 4 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 6 = 4 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 6 = 4 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 6 = 4 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 7 = 5 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000718
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,1,0,0]
 => [1,2] => ([],2)
 => ([],1)
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
Description
The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]
Matching statistic: St001459
(load all 3 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001459: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,1,0,0]
 => [1,2] => ([],2)
 => ([],1)
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
Description
The number of zero columns in the nullspace of a graph.
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000722The number of different neighbourhoods in a graph. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000240The number of indices that are not small excedances. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000868The aid statistic in the sense of Shareshian-Wachs. St000831The number of indices that are either descents or recoils. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000422The energy of a graph, if it is integral. St000141The maximum drop size of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St000264The girth of a graph, which is not a tree. St000474Dyson's crank of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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 $eA$. St001060The distinguishing index of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001875The number of simple modules with projective dimension at most 1. 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. St000672The number of minimal elements in Bruhat order not less than the permutation. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St000742The number of big ascents of a permutation after prepending zero. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001388The number of non-attacking neighbors of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing.