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Your data matches 44 different statistics following compositions of up to 3 maps.
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Matching statistic: St001066
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001483
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St001167
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St000776
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000776: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000776: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [2] => ([],2)
=> ([],1)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(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)
=> 3
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
Description
The maximal multiplicity of an eigenvalue in a graph.
Matching statistic: St000052
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000118
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [.,.]
=> [.,.]
=> 0 = 1 - 1
[1,2] => [.,[.,.]]
=> [[.,.],.]
=> [[.,.],.]
=> 0 = 1 - 1
[2,1] => [[.,.],.]
=> [.,[.,.]]
=> [.,[.,.]]
=> 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [[[.,.],.],.]
=> 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> 0 = 1 - 1
[2,1,3] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0 = 1 - 1
[2,3,1] => [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> 0 = 1 - 1
[3,1,2] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0 = 1 - 1
[3,2,1] => [[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 1 = 2 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> 0 = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> 0 = 1 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> 0 = 1 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> 0 = 1 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> 0 = 1 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0 = 1 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0 = 1 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0 = 1 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0 = 1 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0 = 1 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> 1 = 2 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0 = 1 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 2 = 3 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> 0 = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> 0 = 1 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> 0 = 1 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> 0 = 1 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> 0 = 1 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> 1 = 2 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> 0 = 1 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> 0 = 1 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> 0 = 1 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> 0 = 1 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> 0 = 1 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> 0 = 1 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> 0 = 1 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> 1 = 2 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> 0 = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree.
[[oeis:A001006]] counts binary trees avoiding this pattern.
Matching statistic: St001172
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St000931
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 1 - 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000731
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 0 = 1 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1 = 2 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0 = 1 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 2 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0 = 1 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 2 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 2 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 2 = 3 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[2,1,3,4,5,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 1 - 1
[2,1,3,4,6,5,7] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 1 - 1
[2,1,3,4,6,7,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 1 - 1
[2,1,3,4,7,5,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 1 - 1
[2,1,3,4,7,6,5] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2 - 1
[2,1,3,5,4,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 1 - 1
[2,1,3,5,4,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,3,5,6,4,7] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 1 - 1
[2,1,3,5,6,7,4] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 1 - 1
[2,1,3,5,7,4,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 1 - 1
[2,1,3,5,7,6,4] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2 - 1
[2,1,3,6,4,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 1 - 1
[2,1,3,6,4,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,3,6,5,4,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 2 - 1
[2,1,3,6,5,7,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,3,6,7,4,5] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 1 - 1
[2,1,3,6,7,5,4] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2 - 1
[2,1,3,7,4,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 1 - 1
[2,1,3,7,4,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,3,7,5,4,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 2 - 1
[2,1,3,7,5,6,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,3,7,6,4,5] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 2 - 1
[2,1,3,7,6,5,4] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => ? = 3 - 1
[2,1,4,3,5,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 1 - 1
[2,1,4,3,5,7,6] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 1 - 1
[2,1,4,3,6,5,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 1 - 1
[2,1,4,3,6,7,5] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 1 - 1
[2,1,4,3,7,5,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 1 - 1
[2,1,4,3,7,6,5] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ? = 2 - 1
[2,1,4,5,3,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 1 - 1
[2,1,4,5,3,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,4,5,6,3,7] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 1 - 1
[2,1,4,5,6,7,3] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 1 - 1
[2,1,4,5,7,3,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 1 - 1
[2,1,4,5,7,6,3] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2 - 1
[2,1,4,6,3,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 1 - 1
[2,1,4,6,3,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,4,6,5,3,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 2 - 1
[2,1,4,6,5,7,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,4,6,7,3,5] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 1 - 1
[2,1,4,6,7,5,3] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2 - 1
[2,1,4,7,3,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 1 - 1
[2,1,4,7,3,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,4,7,5,3,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 2 - 1
[2,1,4,7,5,6,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[2,1,4,7,6,3,5] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 2 - 1
[2,1,4,7,6,5,3] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => ? = 3 - 1
[2,1,5,3,4,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 1 - 1
[2,1,5,3,4,7,6] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 1 - 1
[2,1,5,3,6,4,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 1 - 1
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000732The number of double deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000359The number of occurrences of the pattern 23-1. 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. St000039The number of crossings of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation.
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