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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001483
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
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: St001066
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St000931
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 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 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
St000731: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,2,3,6,4,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,2,3,7,4,6] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,6,3,4,7] => ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,7,4] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,2,6,7,3,4] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,6,2,3,4,7] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,3,7,4] => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,4,6] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,6,4,7,5] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,6,7,4,5] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,3,6] => ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,3,5,7] => ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,3,7,5] => ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,7,3,5] => ? = 2 - 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,6,7] => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,6,4,7] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,3,6,7,4] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6] => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,5,6,3,4,7] => ? = 1 - 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,5,6,3,7,4] => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,5,6,7,3,4] => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 1 - 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,3,4,7,5] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,7,4,5] => ? = 1 - 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,4,5,7,6] => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,7] => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,6,7,5] => ? = 3 - 1
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000366
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [3,4,2,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,4,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,5,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [5,2,1,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,3,1,5,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,5,4,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [4,5,3,2,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [3,5,4,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [2,4,5,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [2,5,4,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [2,3,5,4,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [1,7,5,2,3,4,6] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => [1,7,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [5,1,7,4,2,3,6] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => [1,6,5,4,2,3,7] => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => [1,2,7,5,4,3,6] => [1,7,5,4,2,3,6] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [1,2,7,6,4,3,5] => [1,7,6,4,2,3,5] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => [4,1,7,6,2,3,5] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [5,1,2,7,3,4,6] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [1,2,7,6,5,3,4] => [1,7,6,5,2,3,4] => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [1,2,6,3,5,4,7] => [1,6,2,5,3,4,7] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [1,2,7,6,3,5,4] => [7,1,6,5,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [1,2,7,3,5,4,6] => [1,7,2,5,3,4,6] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [1,2,7,3,4,6,5] => [7,1,2,6,3,4,5] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [1,2,7,3,6,4,5] => [1,7,2,6,3,4,5] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [3,1,2,4,7,5,6] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => [7,3,1,2,6,4,5] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => [1,6,3,2,5,4,7] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [1,3,2,7,6,5,4] => [7,6,3,1,5,2,4] => ? = 3 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,4,6] => [1,3,2,7,5,4,6] => [1,7,3,2,5,4,6] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,4,7,5] => [1,3,2,7,6,4,5] => [1,7,3,2,6,4,5] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,4,5] => [1,3,2,7,4,6,5] => [7,1,3,2,6,4,5] => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [1,4,3,2,5,7,6] => [4,1,3,7,5,2,6] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [1,4,3,2,7,6,5] => [4,3,1,7,6,2,5] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [1,5,4,3,2,6,7] => [1,5,6,4,3,2,7] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [1,5,4,3,2,7,6] => [5,1,7,4,3,2,6] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6] => [1,7,5,4,3,2,6] => [1,7,5,4,3,2,6] => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,2,7,5] => [1,7,6,4,3,2,5] => [1,7,6,4,3,2,5] => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,2,5] => [1,7,4,3,2,6,5] => [4,1,7,6,3,2,5] => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => [1,7,4,3,2,5,6] => [1,4,7,5,3,2,6] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,2,4,7,6] => [1,5,3,2,4,7,6] => [5,1,3,7,4,2,6] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,2,6,7,4] => [1,7,6,5,3,2,4] => [1,7,6,5,3,2,4] => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,2,7,4] => [1,7,6,3,2,5,4] => [3,1,7,6,5,2,4] => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,2,4] => [1,7,3,2,6,5,4] => [7,3,1,6,5,2,4] => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,2,7,4,5] => [1,7,3,2,4,6,5] => [3,1,2,7,6,4,5] => ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,2,3,5,7,6] => [1,4,2,3,5,7,6] => [4,1,2,3,7,5,6] => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,2,3,6,7,5] => [1,4,2,3,7,6,5] => [7,4,1,2,6,3,5] => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,2,5,3,7,6] => [1,5,4,2,3,7,6] => [5,1,2,7,4,3,6] => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => [1,7,6,5,4,2,3] => [1,7,6,5,4,2,3] => ? = 4 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,2,6,7,3,5] => [1,7,4,2,3,6,5] => [4,1,2,7,6,3,5] => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,2,3,7,6] => [1,5,2,4,3,7,6] => [5,1,7,2,4,3,6] => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,4,5,2,6,3,7] => [1,6,5,2,4,3,7] => [1,6,2,5,4,3,7] => ? = 2 - 1
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000732
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 83%
Mp00066: Permutations —inverse⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => ? = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,2,6,7,3] => [1,4,7,2,3,5,6] => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,2,7,3] => [1,5,7,2,3,4,6] => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,2,7,3,4] => [1,4,6,7,2,3,5] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => [2,1,3,4,7,5,6] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,5,6] => [2,1,3,4,6,7,5] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => [2,1,3,6,4,5,7] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => [2,1,3,7,4,5,6] => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,4,6] => [2,1,3,6,4,7,5] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => [2,1,3,5,6,4,7] => ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,6,4,7,5] => [2,1,3,5,7,4,6] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,6,7,4,5] => [2,1,3,6,7,4,5] => ? = 1 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => [2,1,3,5,6,7,4] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,5,6] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,6,7,5] => ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,3,4,6,7] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [2,1,5,3,4,7,6] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => [2,1,6,3,4,5,7] => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => [2,1,7,3,4,5,6] => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,3,6] => [2,1,6,3,4,7,5] => ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,3,5,7] => [2,1,5,3,6,4,7] => ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,3,7,5] => [2,1,5,3,7,4,6] => ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,7,3,5] => [2,1,6,3,7,4,5] => ? = 2 - 1
Description
The number of double deficiencies of a permutation.
A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St000365
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [1,2,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => [1,4,2,3] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,4,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => [5,1,2,4,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => [5,1,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [4,5,1,3,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [2,3,4,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [1,3,4,5,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [2,4,5,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,4,5,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,6,5,4,3] => [7,6,1,2,3,4,5] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => [1,2,7,5,4,3,6] => [7,6,1,3,4,5,2] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => [7,6,2,4,5,3,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [1,2,7,6,4,3,5] => [7,6,1,2,4,5,3] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => [7,6,1,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => [7,6,1,4,5,3,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [1,2,6,5,3,4,7] => [7,6,2,3,5,4,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [1,2,7,6,5,3,4] => [7,6,1,2,3,5,4] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [1,2,7,5,3,4,6] => [7,6,1,3,5,4,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [1,2,6,3,5,4,7] => [7,6,2,5,3,4,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [1,2,7,6,3,5,4] => [7,6,1,2,5,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [1,2,7,3,5,4,6] => [7,6,1,5,3,4,2] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [1,2,7,6,3,4,5] => [7,6,1,2,5,4,3] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [1,2,7,3,4,6,5] => [7,6,1,5,4,2,3] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [1,2,7,3,6,4,5] => [7,6,1,5,2,4,3] => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [7,5,6,4,3,1,2] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => [7,5,6,4,1,2,3] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => [7,5,6,4,1,3,2] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => [7,5,6,2,3,4,1] => ? = 2 - 1
Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St000039
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 83%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,7,5,4,6,3] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => ? = 2 - 1
Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000317
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 83%
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => [4,3,2,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => [1,5,2,4,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => [1,5,3,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,5,3,4,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [5,1,2,4,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => [4,1,3,2,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [5,1,3,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => [5,1,4,3,2] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [5,1,3,4,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,2,3,7,4,6,5] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => [1,2,6,3,4,7,5] => [1,2,7,3,4,6,5] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => [1,2,6,3,5,4,7] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [1,2,5,3,7,4,6] => [1,2,7,3,5,4,6] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => [1,2,7,3,6,5,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [1,2,5,3,6,7,4] => [1,2,7,3,5,6,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [1,2,4,6,3,7,5] => [1,2,7,4,3,6,5] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [1,2,6,5,4,3,7] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [1,2,5,7,3,4,6] => [1,2,7,5,4,3,6] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,2,7,6,4,5,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [1,2,5,6,3,7,4] => [1,2,7,5,4,6,3] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,4,5,3,6] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [1,2,4,6,7,3,5] => [1,2,7,4,6,5,3] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [1,2,5,6,7,3,4] => [1,2,7,6,5,4,3] => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,4,5,7] => [1,3,2,6,4,5,7] => ? = 2 - 1
Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Matching statistic: St001189
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 83%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000028The number of stack-sorts needed to sort a permutation. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000665The number of rafts of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000058The order of a permutation. St001862The number of crossings of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001866The nesting alignments of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000485The length of the longest cycle of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001550The number of inversions between exceedances where the greater exceedance is linked.
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