searching the database
Your data matches 101 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001231
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001231: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
Description
The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension.
Actually the same statistics results for algebras with at most 7 simple modules when dropping the assumption that the module has projective dimension one. The author is not sure whether this holds in general.
Matching statistic: St001234
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001234: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
Description
The number of indecomposable three dimensional modules with projective dimension one.
It return zero when there are no such modules.
Matching statistic: St001219
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001219: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001219: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[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]
=> 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]
=> 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
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 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
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[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]
=> 1
[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
[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]
=> 1
[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]
=> 1
[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]
=> 2
[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]
=> 2
[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]
=> 2
[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]
=> 3
[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]
=> 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]
=> 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]
=> 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]
=> 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]
=> 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]
=> 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
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 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]
=> 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]
=> 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
[1,0,1,1,1,0,0,1,0,0]
=> [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
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 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]
=> 1
[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]
=> 1
[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
[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]
=> 1
[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
[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]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
Description
Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive.
Matching statistic: St001210
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,1,0,0]
=> 2 = 0 + 2
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[1,1,0,0]
=> [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[]
=> [] => ?
=> ?
=> ? = 0 + 2
Description
Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001498
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0
[1,1,0,0]
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> ? = 0
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[]
=> [] => ?
=> ?
=> ? = 0
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 01 => 10 => 1 = 0 + 1
[1,0,1,0]
=> 1010 => 0101 => 1010 => 1 = 0 + 1
[1,1,0,0]
=> 1100 => 0011 => 0101 => 2 = 1 + 1
[1,0,1,0,1,0]
=> 101010 => 010101 => 101010 => 1 = 0 + 1
[1,0,1,1,0,0]
=> 101100 => 010011 => 100101 => 1 = 0 + 1
[1,1,0,0,1,0]
=> 110010 => 001101 => 010110 => 2 = 1 + 1
[1,1,0,1,0,0]
=> 110100 => 001011 => 010101 => 2 = 1 + 1
[1,1,1,0,0,0]
=> 111000 => 000111 => 001011 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 10101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 01010011 => 10100101 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 01001101 => 10010110 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 01001011 => 10010101 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 01000111 => 10001011 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 00110101 => 01011010 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 00110011 => 01010101 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 00101101 => 01010110 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 00101011 => 01010101 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 00100111 => 01001011 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 00011101 => 00101110 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 00011011 => 00101101 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 00010111 => 00101011 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 00001111 => 00010111 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 1010101010 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0101010011 => 1010100101 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0101001101 => 1010010110 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0101001011 => 1010010101 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101000111 => 1010001011 => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0100110101 => 1001011010 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0100110011 => 1001010101 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0100101101 => 1001010110 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0100101011 => 1001010101 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0100100111 => 1001001011 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0100011101 => 1000101110 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0100011011 => 1000101101 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0100010111 => 1000101011 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0100001111 => 1000010111 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0011010101 => 0101101010 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0011010011 => 0101100101 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 0011001101 => 0101010110 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0011001011 => 0101010101 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0011000111 => 0101001011 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0010110101 => 0101011010 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0010110011 => 0101010101 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0010101101 => 0101010110 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0010101011 => 0101010101 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0010100111 => 0101001011 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0010011101 => 0100101110 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0010011011 => 0100101101 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0010010111 => 0100101011 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0010001111 => 0100010111 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 0001110101 => 0010111010 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 0001110011 => 0010110101 => 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 0001101101 => 0010110110 => 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0001101011 => 0010110101 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 0001100111 => 0010101011 => 3 = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 0001011101 => 0010101110 => 3 = 2 + 1
[]
=> => => ? => ? = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000806
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 80%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 0 + 2
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [1] => ? = 0 + 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [2] => 3 = 1 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [3] => [1] => ? = 0 + 2
[1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => [1] => ? = 0 + 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => [1,1] => 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1] => 3 = 1 + 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [3] => 4 = 2 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => [1] => ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => [1] => ? = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => [1] => ? = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => [1] => ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1] => ? = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => 3 = 1 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2] => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => [1] => ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [1] => ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => 3 = 1 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => 3 = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => 4 = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => 4 = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => 4 = 2 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => 4 = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => 4 = 2 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => 4 = 2 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => 4 = 2 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => 4 = 2 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => 4 = 2 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => 5 = 3 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => 5 = 3 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => 5 = 3 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => 5 = 3 + 2
[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] => [5] => 6 = 4 + 2
[]
=> ?
=> ? => ? => ? = 0 + 2
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Matching statistic: St000297
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 80%
Mp00104: Binary words —reverse⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> 10 => 01 => 10 => 1 = 0 + 1
[1,0,1,0]
=> 1010 => 0101 => 1000 => 1 = 0 + 1
[1,1,0,0]
=> 1100 => 0011 => 1101 => 2 = 1 + 1
[1,0,1,0,1,0]
=> 101010 => 010101 => 100000 => 1 = 0 + 1
[1,0,1,1,0,0]
=> 101100 => 001101 => 100101 => 1 = 0 + 1
[1,1,0,0,1,0]
=> 110010 => 010011 => 110100 => 2 = 1 + 1
[1,1,0,1,0,0]
=> 110100 => 001011 => 110001 => 2 = 1 + 1
[1,1,1,0,0,0]
=> 111000 => 000111 => 111011 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 10000000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 00110101 => 10000101 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 01001101 => 10010100 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 00101101 => 10010001 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 00011101 => 10011011 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 01010011 => 11010000 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 00110011 => 11010101 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 01001011 => 11000100 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 00101011 => 11000001 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 00011011 => 11001011 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 01000111 => 11101100 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 00100111 => 11101001 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 00010111 => 11100011 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 00001111 => 11110111 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 1000000000 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0011010101 => 1000000101 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0100110101 => 1000010100 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0010110101 => 1000010001 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0001110101 => 1000011011 => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0101001101 => 1001010000 => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0011001101 => 1001010101 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0100101101 => 1001000100 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0010101101 => 1001000001 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0001101101 => 1001001011 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0100011101 => 1001101100 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0010011101 => 1001101001 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0001011101 => 1001100011 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0000111101 => 1001110111 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0101010011 => 1101000000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0011010011 => 1101000101 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 0100110011 => 1101010100 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0010110011 => 1101010001 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0001110011 => 1101011011 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0101001011 => 1100010000 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0011001011 => 1100010101 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0100101011 => 1100000100 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0010101011 => 1100000001 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0001101011 => 1100001011 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0100011011 => 1100101100 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0010011011 => 1100101001 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0001011011 => 1100100011 => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0000111011 => 1100110111 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 0101000111 => 1110110000 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 0011000111 => 1110110101 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 0100100111 => 1110100100 => 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0010100111 => 1110100001 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 0001100111 => 1110101011 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 0100010111 => 1110001100 => 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 0010010111 => 1110001001 => 3 = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 0001010111 => 1110000011 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 0000110111 => 1110010111 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 0100001111 => 1111011100 => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 0010001111 => 1111011001 => ? = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 0001001111 => 1111010011 => ? = 3 + 1
[1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 0000101111 => 1111000111 => ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 0000011111 => 1111101111 => ? = 4 + 1
[]
=> => => => ? = 0 + 1
Description
The number of leading ones in a binary word.
Matching statistic: St001889
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001889: Signed permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001889: Signed permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [-1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [-2,1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,-2] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [-3,1,2] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [-3,2,1] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,-3,2] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1,-3] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,-3] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [-4,1,3,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [-4,2,1,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => [-4,2,3,1] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [-4,3,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,-4,2,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [1,-4,3,2] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [2,1,-4,3] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [3,1,2,-4] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [3,2,1,-4] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,3,2,-4] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [2,1,3,-4] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [-5,1,2,3,4] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,4,3] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [-5,1,3,2,4] => ? = 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => [-5,1,3,4,2] => ? = 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [-5,1,4,3,2] => ? = 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [-5,2,1,3,4] => ? = 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [-5,2,1,4,3] => ? = 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => [-5,2,3,1,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => [-5,2,3,4,1] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,4,2] => [-5,2,4,3,1] => ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [-5,3,2,1,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,5,2] => [-5,3,2,4,1] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,3,2] => [-5,3,4,2,1] => ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [-5,4,3,2,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,-5,2,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,-5,2,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,-5,3,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,5,3] => [1,-5,3,4,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,-5,4,3,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,1,-5,3,4] => ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,5,4] => [2,1,-5,4,3] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [3,1,2,-5,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [4,1,2,3,-5] => ? = 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,5,4,1] => [4,1,3,2,-5] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1,5] => [3,2,1,-5,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,5,1] => [4,2,1,3,-5] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,5,3,1] => [4,2,3,1,-5] => ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [4,3,2,1,-5] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,2,-5,4,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => [1,3,2,-5,4] => 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => [1,4,2,3,-5] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => [1,4,3,2,-5] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => [2,1,3,-5,4] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => [2,1,4,3,-5] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => [3,1,2,4,-5] => ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [3,5,4,2,1] => [3,2,1,4,-5] => ? = 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,2,3,-5,4] => 3
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => [1,2,4,3,-5] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => [1,3,2,4,-5] => 3
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => [2,1,3,4,-5] => ? = 3
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,-5] => 4
[]
=> [] => [] => ? => ? = 0
Description
The size of the connectivity set of a signed permutation.
According to [1], the connectivity set of a signed permutation $w\in\mathfrak H_n$ is $n$ minus the number of generators appearing in any reduced word for $w$.
The connectivity set can be defined for arbitrary Coxeter systems. For permutations, see [[St000234]]. For the number of connected elements in a Coxeter system see [[St001888]].
Matching statistic: St000382
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2] => 2 = 0 + 2
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,2] => 2 = 0 + 2
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [3,1] => 3 = 1 + 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2] => 2 = 0 + 2
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,3,1] => 2 = 0 + 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,1,2] => 3 = 1 + 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1] => 3 = 1 + 2
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,1,1] => 4 = 2 + 2
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,2,2,2] => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,2,3,1] => 2 = 0 + 2
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,3,1,2] => 2 = 0 + 2
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,3,2,1] => 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,4,1,1] => 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,1,2,2] => 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [3,1,3,1] => 3 = 1 + 2
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,2,1,2] => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [3,2,2,1] => 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [3,3,1,1] => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,1,1,2] => 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,1,2,1] => 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,2,1,1] => 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [5,1,1,1] => 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,2,2,2,2] => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,2,2,3,1] => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,2,3,1,2] => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,2,3,2,1] => 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,2,4,1,1] => ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,3,1,2,2] => ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,3,1,3,1] => ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,3,2,1,2] => ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,3,2,2,1] => 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,3,3,1,1] => ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,4,1,1,2] => ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,4,1,2,1] => ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,4,2,1,1] => ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,5,1,1,1] => ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [3,1,2,2,2] => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [3,1,2,3,1] => ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [3,1,3,1,2] => ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [3,1,3,2,1] => ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [3,1,4,1,1] => ? = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [3,2,1,2,2] => ? = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [3,2,1,3,1] => ? = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [3,2,2,1,2] => ? = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => [3,2,2,2,1] => 3 = 1 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => [3,2,3,1,1] => ? = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [3,3,1,1,2] => ? = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => [3,3,1,2,1] => ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [3,3,2,1,1] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => [3,4,1,1,1] => ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [4,1,1,2,2] => ? = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [4,1,1,3,1] => ? = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [4,1,2,1,2] => ? = 2 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => [4,1,2,2,1] => ? = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => [4,1,3,1,1] => ? = 2 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [4,2,1,1,2] => ? = 2 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [4,2,1,2,1] => ? = 2 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => [4,2,2,1,1] => 4 = 2 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [4,3,1,1,1] => 4 = 2 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [5,1,1,1,2] => ? = 3 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => [5,1,1,2,1] => ? = 3 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [5,1,2,1,1] => ? = 3 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => [5,2,1,1,1] => 5 = 3 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => [6,1,1,1,1] => 6 = 4 + 2
[]
=> []
=> ? => ? => ? = 0 + 2
Description
The first part of an integer composition.
The following 91 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St000383The last part of an integer composition. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the permutation. St000338The number of pixed points of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000505The biggest entry in the block containing the 1. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000971The smallest closer of a set partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001937The size of the center of a parking function. St001557The number of inversions of the second entry of a permutation. St000942The number of critical left to right maxima of the parking functions. St001904The length of the initial strictly increasing segment of a parking function. St000439The position of the first down step of a Dyck path. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001645The pebbling number of a connected graph. St000492The rob statistic of a set partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001732The number of peaks visible from the left. St000839The largest opener of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000441The number of successions of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000007The number of saliances of the permutation. St001115The number of even descents of a permutation. St001556The number of inversions of the third entry of a permutation. St001822The number of alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001948The number of augmented double ascents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001096The size of the overlap set of a permutation. St000068The number of minimal elements in a poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000039The number of crossings of a permutation. St000091The descent variation of a composition. St000234The number of global ascents of a permutation. St000317The cycle descent number of a permutation. St000365The number of double ascents of a permutation. St000663The number of right floats of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000989The number of final rises of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001549The number of restricted non-inversions between exceedances. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001781The interlacing number of a set partition. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000051The size of the left subtree of a binary tree. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000314The number of left-to-right-maxima of a permutation. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000729The minimal arc length of a set partition. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001468The smallest fixpoint of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000469The distinguishing number of a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St001366The maximal multiplicity of a degree of a vertex of a graph. St001481The minimal height of a peak of a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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