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Your data matches 60 different statistics following compositions of up to 3 maps.
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Matching statistic: St000394
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000394: 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]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001033
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 98%●distinct values known / distinct values provided: 91%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 98%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 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]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[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
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 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]
=> 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
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[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
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[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]
=> 0
[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
[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]
=> 0
[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]
=> 0
[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]
=> 0
[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]
=> 0
[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
[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
[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
[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]
=> 2
[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]
=> 2
[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]
=> 0
[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]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[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]
=> 0
[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]
=> 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]
=> 0
[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]
=> 0
[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]
=> 0
[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]
=> 0
[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]
=> 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]
=> 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]
=> 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
[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
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 10
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St000496
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1,0]
=> {{1}}
=> {{1}}
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> {{1,6,7,8},{2,5},{3,4}}
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ? = 8
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> {{1,2,4,6,7,8},{3},{5}}
=> ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,6},{5}}
=> {{1,2,6,7,8},{3,5},{4}}
=> ? = 7
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,8},{5},{6}}
=> {{1,2,5,6,7,8},{3},{4}}
=> ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,8,9},{7}}
=> {{1,2,4,5,6,7,8,9},{3}}
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,9},{7,8}}
=> {{1,4,5,6,7,8,9},{2,3}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9,10},{8}}
=> {{1,2,4,5,6,7,8,9,10},{3}}
=> ? = 7
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,8,9},{6},{7}}
=> {{1,2,5,6,7,8,9},{3},{4}}
=> ? = 10
[]
=> []
=> {}
=> {}
=> ? = 0
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,9},{8}}
=> {{1,3,4,5,6,7,8},{2},{9}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> {{1},{2,4,5,6,7,8,9},{3}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9},{8}}
=> {{1,3,4,5,6,7,8,9},{2}}
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> {{1,9},{2,3,4,5,6},{7},{8}}
=> ?
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8,9}}
=> {{1,2,3,4,5},{6,7,8,9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8,9}}
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4},{5,6,7,8,9}}
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8,9}}
=> {{1,2,3,4,5},{6},{7,8,9}}
=> ? = 0
Description
The rcs statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St000491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1,0]
=> {{1}}
=> {{1}}
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> {{1,6,7,8},{2,5},{3,4}}
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ? = 8
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> {{1,2,4,6,7,8},{3},{5}}
=> ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,6},{5}}
=> {{1,2,6,7,8},{3,5},{4}}
=> ? = 7
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,8},{5},{6}}
=> {{1,2,5,6,7,8},{3},{4}}
=> ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,8,9},{7}}
=> {{1,2,4,5,6,7,8,9},{3}}
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,9},{7,8}}
=> {{1,4,5,6,7,8,9},{2,3}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9,10},{8}}
=> {{1,2,4,5,6,7,8,9,10},{3}}
=> ? = 7
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,8,9},{6},{7}}
=> {{1,2,5,6,7,8,9},{3},{4}}
=> ? = 10
[]
=> []
=> {}
=> {}
=> ? = 0
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,9},{8}}
=> {{1,3,4,5,6,7,8},{2},{9}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> {{1},{2,4,5,6,7,8,9},{3}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9},{8}}
=> {{1,3,4,5,6,7,8,9},{2}}
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> {{1},{2,3,4,5,6,7,8,9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> {{1,9},{2,3,4,5,6},{7},{8}}
=> ?
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8,9}}
=> {{1,2,3,4,5},{6,7,8,9}}
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9}}
=> {{1,2,3,4,5,6,7,8},{9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8,9}}
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4},{5,6,7,8,9}}
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8,9}}
=> {{1,2,3,4,5},{6},{7,8,9}}
=> ? = 0
Description
The number of inversions of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$.
This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller".
This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000581
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000581: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000581: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1,0]
=> {{1}}
=> {{1}}
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> {{1,6,7,8},{2,5},{3,4}}
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ? = 8
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> {{1,2,4,6,7,8},{3},{5}}
=> ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,6},{5}}
=> {{1,2,6,7,8},{3,5},{4}}
=> ? = 7
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,8},{5},{6}}
=> {{1,2,5,6,7,8},{3},{4}}
=> ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,8,9},{7}}
=> {{1,2,4,5,6,7,8,9},{3}}
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,9},{7,8}}
=> {{1,4,5,6,7,8,9},{2,3}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9,10},{8}}
=> {{1,2,4,5,6,7,8,9,10},{3}}
=> ? = 7
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,8,9},{6},{7}}
=> {{1,2,5,6,7,8,9},{3},{4}}
=> ? = 10
[]
=> []
=> {}
=> {}
=> ? = 0
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,9},{8}}
=> {{1,3,4,5,6,7,8},{2},{9}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> {{1},{2,4,5,6,7,8,9},{3}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9},{8}}
=> {{1,3,4,5,6,7,8,9},{2}}
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> {{1},{2,3,4,5,6,7,8,9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> {{1,9},{2,3,4,5,6},{7},{8}}
=> ?
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8,9}}
=> {{1,2,3,4,5},{6,7,8,9}}
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9}}
=> {{1,2,3,4,5,6,7,8},{9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8,9}}
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4},{5,6,7,8,9}}
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8,9}}
=> {{1,2,3,4,5},{6},{7,8,9}}
=> ? = 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal.
Matching statistic: St000572
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000572: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 94%●distinct values known / distinct values provided: 91%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000572: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 94%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1,0]
=> [1] => {{1}}
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => {{1,3},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 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] => {{1,2},{3,4},{5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 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] => {{1},{2,3},{4,5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7,8] => ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7,8] => ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7,8] => {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => {{1,5},{2,7},{3},{4},{6},{8}}
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => {{1,6},{2,7},{3},{4},{5},{8}}
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 5
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => {{1},{2,6},{3,8},{4},{5},{7}}
=> ? = 7
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => {{1},{2,7},{3,8},{4},{5},{6}}
=> ? = 8
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 5
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => {{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => {{1,8},{2},{3},{4},{5},{6},{7},{9}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,2,3,4,5,6,7,8,9] => {{1},{2,10},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 7
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => {{1},{2,8},{3,9},{4},{5},{6},{7}}
=> ? = 10
[]
=> []
=> [] => ?
=> ? = 0
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => {{1,2,9},{3},{4},{5},{6},{7},{8}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,1,9] => {{1,2,3,4,5,6,7,8},{9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,4,5,1,2,6,7,8,9] => ?
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8,9] => ?
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,5] => {{1},{2},{3},{4},{5,6,7,8,9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => {{1},{2,3,4,5,6,7,8,9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,8,5,9] => ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5,8,9] => ?
=> ? = 0
Description
The dimension exponent of a set partition.
This is
$$\sum_{B\in\pi} (\max(B) - \min(B) + 1) - n$$
where the summation runs over the blocks of the set partition $\pi$ of $\{1,\dots,n\}$.
It is thus equal to the difference [[St000728]] - [[St000211]].
This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 and 3 are consecutive elements in a block.
This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 is the minimal and 3 is the maximal element of the block.
Matching statistic: St000497
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 94%●distinct values known / distinct values provided: 91%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 94%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => {{1}}
=> {{1}}
=> ? = 0
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> {{1},{2}}
=> 0
[1,1,0,0]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> {{1},{2,3}}
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
[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
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> {{1},{2},{3,5},{4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> {{1},{2,4,5},{3}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1},{2,5},{3,4}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> {{1},{2,5},{3},{4}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => {{1},{2},{3},{4},{5},{6,7,8}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8] => ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => {{1},{2},{3},{4},{5,6,7,8}}
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => {{1},{2,7},{3},{4,8},{5},{6}}
=> {{1},{2,8},{3},{4,7},{5},{6}}
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => {{1},{2,7},{3,8},{4},{5},{6}}
=> {{1},{2,8},{3,7},{4},{5},{6}}
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ?
=> ?
=> ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => {{1,2,8},{3},{4},{5},{6},{7}}
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,6,7,8,2] => ?
=> ?
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => {{1,5},{2,7},{3},{4},{6},{8}}
=> ?
=> ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,7,2,3,4,5,8] => {{1,6},{2},{3,7},{4},{5},{8}}
=> ?
=> ? = 7
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => {{1,6},{2,7},{3},{4},{5},{8}}
=> ?
=> ? = 8
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 5
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => {{1,7,8},{2},{3},{4},{5},{6}}
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => {{1,8},{2},{3},{4},{5},{6},{7},{9}}
=> {{1,8},{2},{3},{4},{5},{6},{7},{9}}
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => {{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> {{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => {{1,9},{2},{3},{4},{5},{6},{7},{8},{10}}
=> {{1,9},{2},{3},{4},{5},{6},{7},{8},{10}}
=> ? = 7
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,1,2,3,4,5,6,9] => {{1,7},{2,8},{3},{4},{5},{6},{9}}
=> ?
=> ? = 10
[]
=> [] => ?
=> ?
=> ? = 0
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => {{1,2,9},{3},{4},{5},{6},{7},{8}}
=> {{1,2,9},{3},{4},{5},{6},{7},{8}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => {{1},{2,3,4,5,6,7,8,9}}
=> {{1},{2,3,4,5,6,7,8,9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,4,7,8,9,5,6] => ?
=> ?
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1,6,7,8,9] => ?
=> ?
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,1,9] => {{1,2,3,4,5,6,7,8},{9}}
=> {{1,2,3,4,5,6,7,8},{9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,2,6,7,8,9] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8,9] => ?
=> ?
=> ? = 0
Description
The lcb statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Matching statistic: St000609
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 94%●distinct values known / distinct values provided: 91%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 94%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => {{1}}
=> {{1}}
=> ? = 0
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> {{1},{2}}
=> 0
[1,1,0,0]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => {{1,3},{2}}
=> {{1},{2,3}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 2
[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
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1},{2,4,5},{3}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => {{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8] => ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => {{1},{2},{3},{4},{5,6,7,8}}
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ?
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ?
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => {{1},{2,7},{3,8},{4},{5},{6}}
=> {{1},{2},{3},{4},{5,7,8},{6}}
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ?
=> ?
=> ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => {{1,2,8},{3},{4},{5},{6},{7}}
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,6,7,8,2] => ?
=> ?
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => {{1,5},{2,7},{3},{4},{6},{8}}
=> ?
=> ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,7,2,3,4,5,8] => {{1,6},{2},{3,7},{4},{5},{8}}
=> ?
=> ? = 7
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => {{1,6},{2,7},{3},{4},{5},{8}}
=> ?
=> ? = 8
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 5
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => {{1,7,8},{2},{3},{4},{5},{6}}
=> {{1,8},{2},{3},{4},{5},{6,7}}
=> ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => {{1,8},{2},{3},{4},{5},{6},{7},{9}}
=> {{1},{2},{3},{4},{5},{6},{7,8},{9}}
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => {{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7,9},{8}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => {{1,9},{2},{3},{4},{5},{6},{7},{8},{10}}
=> {{1},{2},{3},{4},{5},{6},{7},{8,9},{10}}
=> ? = 7
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,1,2,3,4,5,6,9] => {{1,7},{2,8},{3},{4},{5},{6},{9}}
=> ?
=> ? = 10
[]
=> [] => ?
=> ?
=> ? = 0
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => {{1,2,9},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> {{1,9},{2},{3},{4},{5},{6},{7,8}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => {{1},{2,3,4,5,6,7,8,9}}
=> {{1,3,4,5,6,7,8,9},{2}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,4,7,8,9,5,6] => ?
=> ?
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1,6,7,8,9] => ?
=> ?
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,1,9] => {{1,2,3,4,5,6,7,8},{9}}
=> {{1,2,3,4,5,6,7,8},{9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,2,6,7,8,9] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8,9] => ?
=> ?
=> ? = 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St001843
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001843: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 94%●distinct values known / distinct values provided: 91%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001843: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 94%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1,0]
=> {{1}}
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6},{7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6,7}}
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8}}
=> ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,6},{5}}
=> ? = 7
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 8
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,7,8},{6}}
=> ? = 5
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7},{6},{8}}
=> ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,8},{7}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,8,9},{7}}
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,9},{7,8}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9,10},{8}}
=> ? = 7
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,8,9},{6},{7}}
=> ? = 10
[]
=> []
=> {}
=> ? = 0
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,9},{8}}
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9},{8}}
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> {{1,9},{2,3,4,5,6},{7},{8}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8,9}}
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8,9}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4},{5,6,7,8,9}}
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8,9}}
=> ? = 0
Description
The Z-index of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
The Z-index of $w$ equals
$$
\sum_{i < j} w_{i,j},
$$
where $w_{i,j}$ is the word obtained from $w$ by removing all letters different from $i$ and $j$.
Matching statistic: St000065
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 91%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [[1]]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 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]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 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]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[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]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[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]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[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]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 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]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[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]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[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]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 0
[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]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[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]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 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]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 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]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 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]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 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]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 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]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 2
[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]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 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]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 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]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 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]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 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]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[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]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
[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]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
[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]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2
[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]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 3
[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]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,1,0,-1,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2
Description
The number of entries equal to -1 in an alternating sign matrix.
The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000356The number of occurrences of the pattern 13-2. St001727The number of invisible inversions of a permutation. St000359The number of occurrences of the pattern 23-1. St000358The number of occurrences of the pattern 31-2. St000223The number of nestings in the permutation. St000355The number of occurrences of the pattern 21-3. St001083The number of boxed occurrences of 132 in a permutation. St000039The number of crossings of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001725The harmonious chromatic number of a graph. St000456The monochromatic index of a connected graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001811The Castelnuovo-Mumford regularity of a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001866The nesting alignments of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001862The number of crossings of a signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001905The number of preferred parking spots in a parking function less than the index of the car.
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