Your data matches 76 different statistics following compositions of up to 3 maps.
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Matching statistic: St000757
(load all 2 compositions to match this statistic)
Mapping
St000757: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 2
[2,1] => 1
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 3
[1,2,1] => 2
[1,3] => 2
[2,1,1] => 2
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 4
[1,1,2,1] => 3
[1,1,3] => 3
[1,2,1,1] => 3
[1,2,2] => 3
[1,3,1] => 2
[1,4] => 2
[2,1,1,1] => 3
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 2
[3,1,1] => 2
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 5
[1,1,1,2,1] => 4
[1,1,1,3] => 4
[1,1,2,1,1] => 4
[1,1,2,2] => 4
[1,1,3,1] => 3
[1,1,4] => 3
[1,2,1,1,1] => 4
[1,2,1,2] => 3
[1,2,2,1] => 3
[1,2,3] => 3
[1,3,1,1] => 3
[1,3,2] => 2
[1,4,1] => 2
[1,5] => 2
[2,1,1,1,1] => 4
[2,1,1,2] => 3
[2,1,2,1] => 2
Description
The length of the longest weakly inreasing subsequence of parts of an integer composition.
Matching statistic: St001024
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => [1,0]
 => 1
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[3,1] => [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
[4] => [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
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,1,2] => [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]
 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,1,3] => [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]
 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,2,2] => [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]
 => 2
[1,3,1] => [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]
 => 2
[1,4] => [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]
 => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[2,3] => [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
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001733
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => [1,0]
 => 1
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,1] => [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
[4] => [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
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,1,2] => [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]
 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,1,3] => [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]
 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,2,2] => [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]
 => 3
[1,3,1] => [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]
 => 2
[1,4] => [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]
 => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 4
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
Description
The number of weak left to right maxima of a Dyck path. A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its left.
Matching statistic: St000120
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 90%●distinct values known / distinct values provided: 83%
Values
[1] => [[1],[]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1] => [[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2] => [[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => 2
[1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,2] => [[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
[2,1] => [[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3] => [[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,3] => [[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[2,2] => [[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[3,1] => [[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[4] => [[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 2
[1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[1,4] => [[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[2,3] => [[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[3,1,1] => [[3,3,3],[2,2]]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
[3,2] => [[4,3],[2]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[4,1] => [[4,4],[3]]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
[5] => [[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,1,3,4,5,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,3] => [[3,1,1,1],[]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,3,1] => [[3,3,1,1],[2]]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,1,4] => [[4,1,1],[]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[1,4,1] => [[4,4,1],[3]]
 => [4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => 4
[1,5] => [[5,1],[]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 4
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,3,4,5,6}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,3,4,5,6}
[4,1,1] => [[4,4,4],[3,3]]
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? ∊ {1,1,3,4,5,6}
[5,1] => [[5,5],[4]]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? ∊ {1,1,3,4,5,6}
[6] => [[6],[]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {1,1,3,4,5,6}
Description
The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St000888
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000888: Alternating sign matrices ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => [[1]]
 => ? = 1
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => 2
[2] => [1,1,0,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 2
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [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
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [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
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [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]]
 => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [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]]
 => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [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]]
 => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [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]]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [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]]
 => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [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
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [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]]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [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]]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [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]]
 => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,-1,1],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 3
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 3
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[1,0,-1,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,4}
Description
The maximal sum of entries on a diagonal of an alternating sign matrix. For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)$$ are $(0,1,1,0,1,1,0)$, so the statistic is $1$. This is a natural extension of [[St000887]] to alternating sign matrices.
Matching statistic: St000144
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Values
[1] => [[1],[]]
 => []
 => []
 => ? = 1
[1,1] => [[1,1],[]]
 => []
 => []
 => ? ∊ {1,2}
[2] => [[2],[]]
 => []
 => []
 => ? ∊ {1,2}
[1,1,1] => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,3}
[1,2] => [[2,1],[]]
 => []
 => []
 => ? ∊ {1,2,3}
[2,1] => [[2,2],[1]]
 => [1]
 => [1,0]
 => 1
[3] => [[3],[]]
 => []
 => []
 => ? ∊ {1,2,3}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,3,4}
[1,1,2] => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,3,4}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 1
[1,3] => [[3,1],[]]
 => []
 => []
 => ? ∊ {2,2,3,4}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0]
 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[4] => [[4],[]]
 => []
 => []
 => ? ∊ {2,2,3,4}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {2,3,3,4,5}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {2,3,3,4,5}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[1,1,3] => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {2,3,3,4,5}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,4] => [[4,1],[]]
 => []
 => []
 => ? ∊ {2,3,3,4,5}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 3
[3,2] => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[4,1] => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[5] => [[5],[]]
 => []
 => []
 => ? ∊ {2,3,3,4,5}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,4,4,5,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,4,4,5,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,4,4,5,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,1,4] => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,4,4,5,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 3
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[1,5] => [[5,1],[]]
 => []
 => []
 => ? ∊ {2,2,4,4,5,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 3
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0]
 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 4
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 3
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 4
[3,3] => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
[4,2] => [[5,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[5,1] => [[5,5],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[6] => [[6],[]]
 => []
 => []
 => ? ∊ {2,2,4,4,5,6}
Description
The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St000384
(load all 2 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 83%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? ∊ {1,2}
[2] => [[2],[]]
 => []
 => ?
 => ? ∊ {1,2}
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? ∊ {1,2,3}
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? ∊ {1,2,3}
[2,1] => [[2,2],[1]]
 => [1]
 => [1]
 => 1
[3] => [[3],[]]
 => []
 => ?
 => ? ∊ {1,2,3}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,4}
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,4}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,4}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [2]
 => 2
[2,2] => [[3,2],[1]]
 => [1]
 => [1]
 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => [1,1]
 => 2
[4] => [[4],[]]
 => []
 => ?
 => ? ∊ {2,2,3,4}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,1]
 => 2
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [2,1]
 => 2
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [2]
 => 2
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,1,1]
 => 3
[2,3] => [[4,2],[1]]
 => [1]
 => [1]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [4]
 => 4
[3,2] => [[4,3],[2]]
 => [2]
 => [1,1]
 => 2
[4,1] => [[4,4],[3]]
 => [3]
 => [3]
 => 3
[5] => [[5],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,2,3,4,5}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,2,3,4,5}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,2,3,4,5}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,1]
 => 2
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,2,3,4,5}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [2,1]
 => 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,1,1]
 => 3
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [4]
 => 4
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,1]
 => 2
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [3]
 => 3
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? ∊ {2,2,2,3,4,5}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [2,2]
 => 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [2,1]
 => 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [2,1,1]
 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [2]
 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [4,1]
 => 4
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,1,1]
 => 3
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [3,1]
 => 3
[2,4] => [[5,2],[1]]
 => [1]
 => [1]
 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [4,1,1]
 => 4
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [4]
 => 4
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [3,1,1]
 => 3
[3,3] => [[5,3],[2]]
 => [2]
 => [1,1]
 => 2
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [6]
 => 6
[4,2] => [[5,4],[3]]
 => [3]
 => [3]
 => 3
[5,1] => [[5,5],[4]]
 => [4]
 => [1,1,1,1]
 => 4
[6] => [[6],[]]
 => []
 => ?
 => ? ∊ {2,2,2,3,4,5}
Description
The maximal part of the shifted composition of an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part. The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$. See also [[St000380]].
Matching statistic: St000784
(load all 2 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 83%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? ∊ {1,2}
[2] => [[2],[]]
 => []
 => ?
 => ? ∊ {1,2}
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? ∊ {1,2,3}
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? ∊ {1,2,3}
[2,1] => [[2,2],[1]]
 => [1]
 => [1]
 => 1
[3] => [[3],[]]
 => []
 => ?
 => ? ∊ {1,2,3}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,4}
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,4}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,4}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [2]
 => 2
[2,2] => [[3,2],[1]]
 => [1]
 => [1]
 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => [1,1]
 => 2
[4] => [[4],[]]
 => []
 => ?
 => ? ∊ {2,2,3,4}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,1]
 => 2
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [2,1]
 => 2
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [2]
 => 2
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,1,1]
 => 3
[2,3] => [[4,2],[1]]
 => [1]
 => [1]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [4]
 => 4
[3,2] => [[4,3],[2]]
 => [2]
 => [1,1]
 => 2
[4,1] => [[4,4],[3]]
 => [3]
 => [3]
 => 3
[5] => [[5],[]]
 => []
 => ?
 => ? ∊ {2,3,3,3,5}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,3,4,5}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,3,4,5}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,3,4,5}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,1]
 => 2
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,3,4,5}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [2,1]
 => 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,1,1]
 => 3
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [4]
 => 4
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,1]
 => 2
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [3]
 => 3
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? ∊ {2,2,3,3,4,5}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [2,2]
 => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [2,1]
 => 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [2,1,1]
 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [2]
 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [4,1]
 => 4
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,1,1]
 => 3
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [3,1]
 => 3
[2,4] => [[5,2],[1]]
 => [1]
 => [1]
 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [4,1,1]
 => 4
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [4]
 => 4
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [3,1,1]
 => 3
[3,3] => [[5,3],[2]]
 => [2]
 => [1,1]
 => 2
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [6]
 => 6
[4,2] => [[5,4],[3]]
 => [3]
 => [3]
 => 3
[5,1] => [[5,5],[4]]
 => [4]
 => [1,1,1,1]
 => 4
[6] => [[6],[]]
 => []
 => ?
 => ? ∊ {2,2,3,3,4,5}
Description
The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].
Matching statistic: St000969
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Values
[1] => [[1],[]]
 => []
 => []
 => ? = 1
[1,1] => [[1,1],[]]
 => []
 => []
 => ? ∊ {1,2}
[2] => [[2],[]]
 => []
 => []
 => ? ∊ {1,2}
[1,1,1] => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,3}
[1,2] => [[2,1],[]]
 => []
 => []
 => ? ∊ {1,1,3}
[2,1] => [[2,2],[1]]
 => [1]
 => [1,0]
 => 2
[3] => [[3],[]]
 => []
 => []
 => ? ∊ {1,1,3}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,2,4}
[1,1,2] => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,2,4}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,3] => [[3,1],[]]
 => []
 => []
 => ? ∊ {1,1,2,4}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0]
 => 2
[3,1] => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[4] => [[4],[]]
 => []
 => []
 => ? ∊ {1,1,2,4}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,3,5}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,3,5}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,3] => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,3,5}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,4] => [[4,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,3,5}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0]
 => 2
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[3,2] => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[4,1] => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 4
[5] => [[5],[]]
 => []
 => []
 => ? ∊ {1,1,1,3,5}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,4,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,4,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,4,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,4] => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,4,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 4
[1,5] => [[5,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,4,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0]
 => 2
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 2
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
[3,3] => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 3
[4,2] => [[5,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 4
[5,1] => [[5,5],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 5
[6] => [[6],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,4,6}
Description
We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. Then we calculate the global dimension of that CNakayama algebra.
Matching statistic: St001570
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ? = 1
[1,1] => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,2}
[2] => ([],2)
 => ([],2)
 => ? ∊ {1,2}
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {1,2}
[1,2] => ([(1,2)],3)
 => ([],2)
 => ? ∊ {1,2}
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[3] => ([],3)
 => ([],3)
 => 3
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,2,2}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {1,2,2}
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => 3
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,2,2}
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[4] => ([],4)
 => ([],4)
 => 4
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,2,2,2,2,3}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {1,2,2,2,2,3}
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 3
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,2,2,2,2,3}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 4
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,2,2,2,2,3}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {1,2,2,2,2,3}
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 3
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,2,2,2,2,3}
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[5] => ([],5)
 => ([],5)
 => 5
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 3
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 4
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,4),(3,4)],5)
 => 3
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,5] => ([(4,5)],6)
 => ([],5)
 => 5
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 3
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4] => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => 4
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => 4
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[6] => ([],6)
 => ([],6)
 => 6
Description
The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
The following 66 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000028The number of stack-sorts needed to sort a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000335The difference of lower and upper interactions. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001118The acyclic chromatic index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000741The Colin de Verdière graph invariant. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St000454The largest eigenvalue of a graph if it is integral. St001615The number of join prime elements of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St001948The number of augmented double ascents of a permutation. St000365The number of double ascents of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001730The number of times the path corresponding to a binary word crosses the base line. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000352The Elizalde-Pak rank of a permutation. St001096The size of the overlap set of a permutation. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000090The variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000663The number of right floats of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001556The number of inversions of the third entry of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.