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Your data matches 86 different statistics following compositions of up to 3 maps.
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Matching statistic: St000676
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[2] => [1] => [1,0]
=> 1
[1,2] => [1,1] => [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> 2
[3] => [1] => [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3] => [1,1] => [1,0,1,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> 2
[4] => [1] => [1,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,4] => [1,1] => [1,0,1,0]
=> 2
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3] => [1,1] => [1,0,1,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> 2
[4,1] => [1,1] => [1,0,1,0]
=> 2
[5] => [1] => [1,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,5] => [1,1] => [1,0,1,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,4] => [1,1] => [1,0,1,0]
=> 2
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[4,2] => [1,1] => [1,0,1,0]
=> 2
[5,1] => [1,1] => [1,0,1,0]
=> 2
[6] => [1] => [1,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,6] => [1,1] => [1,0,1,0]
=> 2
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,5] => [1,1] => [1,0,1,0]
=> 2
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001183
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001183: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001183: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[2] => [1] => [1,0]
=> 1
[1,2] => [1,1] => [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> 2
[3] => [1] => [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3] => [1,1] => [1,0,1,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> 2
[4] => [1] => [1,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,4] => [1,1] => [1,0,1,0]
=> 2
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3] => [1,1] => [1,0,1,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> 2
[4,1] => [1,1] => [1,0,1,0]
=> 2
[5] => [1] => [1,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,5] => [1,1] => [1,0,1,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,4] => [1,1] => [1,0,1,0]
=> 2
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[4,2] => [1,1] => [1,0,1,0]
=> 2
[5,1] => [1,1] => [1,0,1,0]
=> 2
[6] => [1] => [1,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,6] => [1,1] => [1,0,1,0]
=> 2
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,5] => [1,1] => [1,0,1,0]
=> 2
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
Description
The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001258
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001258: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001258: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[2] => [1] => [1,0]
=> 1
[1,2] => [1,1] => [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> 2
[3] => [1] => [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3] => [1,1] => [1,0,1,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> 2
[4] => [1] => [1,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,4] => [1,1] => [1,0,1,0]
=> 2
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3] => [1,1] => [1,0,1,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> 2
[4,1] => [1,1] => [1,0,1,0]
=> 2
[5] => [1] => [1,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,5] => [1,1] => [1,0,1,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,4] => [1,1] => [1,0,1,0]
=> 2
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[4,2] => [1,1] => [1,0,1,0]
=> 2
[5,1] => [1,1] => [1,0,1,0]
=> 2
[6] => [1] => [1,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,6] => [1,1] => [1,0,1,0]
=> 2
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,5] => [1,1] => [1,0,1,0]
=> 2
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.
For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
Matching statistic: St000741
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0 = 1 - 1
[2] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,1] => [4] => ([],4)
=> 1 = 2 - 1
[1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[4] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,4] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,3] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[4,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[5] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,5] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,4] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[4,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[5,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[6] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,4] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,1,2,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,1,3] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,3,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,5,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,6] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,5] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St000010
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 1
[2] => [1] => [1]
=> [1]
=> 1
[1,2] => [1,1] => [1,1]
=> [1,1]
=> 2
[2,1] => [1,1] => [1,1]
=> [1,1]
=> 2
[3] => [1] => [1]
=> [1]
=> 1
[1,1,1,1] => [4] => [4]
=> [2,2]
=> 2
[1,2,1] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[1,3] => [1,1] => [1,1]
=> [1,1]
=> 2
[3,1] => [1,1] => [1,1]
=> [1,1]
=> 2
[4] => [1] => [1]
=> [1]
=> 1
[1,1,1,2] => [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,3,1] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[1,4] => [1,1] => [1,1]
=> [1,1]
=> 2
[2,1,2] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[2,3] => [1,1] => [1,1]
=> [1,1]
=> 2
[3,2] => [1,1] => [1,1]
=> [1,1]
=> 2
[4,1] => [1,1] => [1,1]
=> [1,1]
=> 2
[5] => [1] => [1]
=> [1]
=> 1
[1,1,1,3] => [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[1,3,2] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[1,4,1] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[1,5] => [1,1] => [1,1]
=> [1,1]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 4
[2,1,3] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[2,3,1] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[2,4] => [1,1] => [1,1]
=> [1,1]
=> 2
[3,1,2] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[3,2,1] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[4,2] => [1,1] => [1,1]
=> [1,1]
=> 2
[5,1] => [1,1] => [1,1]
=> [1,1]
=> 2
[6] => [1] => [1]
=> [1]
=> 1
[1,1,1,4] => [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,2,1,2,1] => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[1,2,1,3] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,1] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,4] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[1,3,1,2] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 4
[1,3,2,1] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 4
[1,4,2] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[1,5,1] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[1,6] => [1,1] => [1,1]
=> [1,1]
=> 2
[2,1,3,1] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 4
[2,1,4] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[2,2,2,1] => [3,1] => [3,1]
=> [2,1,1]
=> 3
[2,3,2] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[2,4,1] => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 3
[2,5] => [1,1] => [1,1]
=> [1,1]
=> 2
[3,1,2,1] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000013
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[2] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[4] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,4] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,3] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[4,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[5] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,5] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,4] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[4,2] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[5,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[6] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,6] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,5] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000031
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 1
[2] => [1] => [1,0]
=> [1] => 1
[1,2] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3] => [1] => [1,0]
=> [1] => 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3,1] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[4] => [1] => [1,0]
=> [1] => 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,4] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,3] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3,2] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[4,1] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[5] => [1] => [1,0]
=> [1] => 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,5] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,4] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[4,2] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[5,1] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[6] => [1] => [1,0]
=> [1] => 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,6] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,5] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000144
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[2] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[4] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,4] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,3] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[4,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[5] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,5] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,4] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[4,2] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[5,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[6] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,6] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,5] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
Description
The pyramid weight of the Dyck path.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St000213
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 1
[2] => [1] => [1,0]
=> [1] => 1
[1,2] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3] => [1] => [1,0]
=> [1] => 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3,1] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[4] => [1] => [1,0]
=> [1] => 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,4] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,3] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3,2] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[4,1] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[5] => [1] => [1,0]
=> [1] => 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,5] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,4] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[4,2] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[5,1] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[6] => [1] => [1,0]
=> [1] => 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,6] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,5] => [1,1] => [1,0,1,0]
=> [1,2] => 2
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000288
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 1 => 1
[2] => [1] => 1 => 1 => 1
[1,2] => [1,1] => 11 => 11 => 2
[2,1] => [1,1] => 11 => 11 => 2
[3] => [1] => 1 => 1 => 1
[1,1,1,1] => [4] => 1000 => 0101 => 2
[1,2,1] => [1,1,1] => 111 => 111 => 3
[1,3] => [1,1] => 11 => 11 => 2
[3,1] => [1,1] => 11 => 11 => 2
[4] => [1] => 1 => 1 => 1
[1,1,1,2] => [3,1] => 1001 => 1101 => 3
[1,3,1] => [1,1,1] => 111 => 111 => 3
[1,4] => [1,1] => 11 => 11 => 2
[2,1,2] => [1,1,1] => 111 => 111 => 3
[2,3] => [1,1] => 11 => 11 => 2
[3,2] => [1,1] => 11 => 11 => 2
[4,1] => [1,1] => 11 => 11 => 2
[5] => [1] => 1 => 1 => 1
[1,1,1,3] => [3,1] => 1001 => 1101 => 3
[1,2,1,2] => [1,1,1,1] => 1111 => 1111 => 4
[1,2,3] => [1,1,1] => 111 => 111 => 3
[1,3,2] => [1,1,1] => 111 => 111 => 3
[1,4,1] => [1,1,1] => 111 => 111 => 3
[1,5] => [1,1] => 11 => 11 => 2
[2,1,2,1] => [1,1,1,1] => 1111 => 1111 => 4
[2,1,3] => [1,1,1] => 111 => 111 => 3
[2,3,1] => [1,1,1] => 111 => 111 => 3
[2,4] => [1,1] => 11 => 11 => 2
[3,1,2] => [1,1,1] => 111 => 111 => 3
[3,2,1] => [1,1,1] => 111 => 111 => 3
[4,2] => [1,1] => 11 => 11 => 2
[5,1] => [1,1] => 11 => 11 => 2
[6] => [1] => 1 => 1 => 1
[1,1,1,4] => [3,1] => 1001 => 1101 => 3
[1,2,1,2,1] => [1,1,1,1,1] => 11111 => 11111 => 5
[1,2,1,3] => [1,1,1,1] => 1111 => 1111 => 4
[1,2,3,1] => [1,1,1,1] => 1111 => 1111 => 4
[1,2,4] => [1,1,1] => 111 => 111 => 3
[1,3,1,2] => [1,1,1,1] => 1111 => 1111 => 4
[1,3,2,1] => [1,1,1,1] => 1111 => 1111 => 4
[1,4,2] => [1,1,1] => 111 => 111 => 3
[1,5,1] => [1,1,1] => 111 => 111 => 3
[1,6] => [1,1] => 11 => 11 => 2
[2,1,3,1] => [1,1,1,1] => 1111 => 1111 => 4
[2,1,4] => [1,1,1] => 111 => 111 => 3
[2,2,2,1] => [3,1] => 1001 => 1101 => 3
[2,3,2] => [1,1,1] => 111 => 111 => 3
[2,4,1] => [1,1,1] => 111 => 111 => 3
[2,5] => [1,1] => 11 => 11 => 2
[3,1,2,1] => [1,1,1,1] => 1111 => 1111 => 4
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 76 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000746The number of pairs with odd minimum in a perfect matching. St000784The maximum of the length and the largest part of the integer partition. St000918The 2-limited packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001007Number of simple modules with projective dimension 1 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. St001119The length of a shortest maximal 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001267The length of the Lyndon factorization of the binary word. St001315The dissociation number of a graph. 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. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001746The coalition number of a graph. St000024The number of double up and double down steps of a Dyck path. St000295The length of the border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000377The dinv defect of an integer partition. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in 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 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001955The number of natural descents for set-valued two row standard Young tableaux. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000675The number of centered multitunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St001644The dimension of a graph. St000060The greater neighbor of the maximum. St000083The number of left oriented leafs of a binary tree except the first one. St000442The maximal area to the right of an up step of a Dyck path. St001246The maximal difference between two consecutive entries of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000836The number of descents of distance 2 of a permutation. 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. St001645The pebbling number of a connected graph. St000438The position of the last up step in a Dyck path. St001280The number of parts of an integer partition that are at least two. St001270The bandwidth of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001962The proper pathwidth of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St000993The multiplicity of the largest part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000782The indicator function of whether a given perfect matching is an L & P matching. St000897The number of different multiplicities of parts of an integer partition. St001512The minimum rank of a graph. St000242The number of indices that are not cyclical small weak excedances. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000624The normalized sum of the minimal distances to a greater element. St001323The independence gap of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph.
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