Your data matches 52 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001237
(load all 175 compositions to match this statistic)
Mapping
St001237: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 2
[1,0,1,0]
 => 3
[1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => 4
[1,1,0,0,1,0]
 => 3
[1,1,0,1,0,0]
 => 4
[1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => 5
[1,0,1,1,1,0,0,0]
 => 5
[1,1,0,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => 4
[1,1,1,0,0,1,0,0]
 => 4
[1,1,1,0,1,0,0,0]
 => 5
[1,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => 6
[1,0,1,1,0,1,1,0,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => 5
[1,0,1,1,1,0,0,1,0,0]
 => 5
[1,0,1,1,1,0,1,0,0,0]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => 6
Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
Matching statistic: St001211
(load all 12 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001211: 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]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 4
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 5
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 6
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 5
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 5
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 6
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St000250
(load all 2 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000250: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => {{1,2}}
 => {{1},{2}}
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 3
[1,1,0,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1},{2},{3}}
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => 3
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 4
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1},{2,3,4,5}}
 => 5
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => {{1},{2},{3,4,5}}
 => 5
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => {{1},{2,4,5},{3}}
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => {{1},{2,3},{4,5}}
 => 5
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => {{1},{2},{3},{4,5}}
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => {{1},{2,3,5},{4}}
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => {{1},{2},{3,5},{4}}
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => {{1},{2,5},{3,4}}
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => {{1},{2,3,4},{5}}
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => {{1},{2},{3,4},{5}}
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => {{1},{2,5},{3},{4}}
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => {{1},{2,4},{3},{5}}
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => {{1},{2,3},{4},{5}}
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,6},{2},{3},{4},{5}}
 => {{1},{2,3,4,5,6}}
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2},{3},{4}}
 => {{1},{2},{3,4,5,6}}
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => {{1,6},{2},{3},{4,5}}
 => {{1},{2,4,5,6},{3}}
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => {{1,4,6},{2},{3},{5}}
 => {{1},{2,3},{4,5,6}}
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2},{3}}
 => {{1},{2},{3},{4,5,6}}
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => {{1,6},{2},{3,4},{5}}
 => {{1},{2,3,5,6},{4}}
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => {{1,5,6},{2},{3,4}}
 => {{1},{2},{3,5,6},{4}}
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => {{1,6},{2},{3,5},{4}}
 => {{1},{2,5,6},{3,4}}
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => {{1,3,6},{2},{4},{5}}
 => {{1},{2,3,4},{5,6}}
 => 6
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => {{1,3,5,6},{2},{4}}
 => {{1},{2},{3,4},{5,6}}
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => {{1,6},{2},{3,4,5}}
 => {{1},{2,5,6},{3},{4}}
 => 5
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => {{1,3,6},{2},{4,5}}
 => {{1},{2,4},{3},{5,6}}
 => 5
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => {{1,3,4,6},{2},{5}}
 => {{1},{2,3},{4},{5,6}}
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => {{1,3,4,5,6},{2}}
 => {{1},{2},{3},{4},{5,6}}
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => {{1},{2,3,4,6},{5}}
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2,3},{4}}
 => {{1},{2},{3,4,6},{5}}
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => {{1,6},{2,3},{4,5}}
 => {{1},{2,4,6},{3},{5}}
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => {{1,4,6},{2,3},{5}}
 => {{1},{2,3},{4,6},{5}}
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2,3}}
 => {{1},{2},{3},{4,6},{5}}
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => {{1,6},{2,4},{3},{5}}
 => {{1},{2,3,6},{4,5}}
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => {{1,5,6},{2,4},{3}}
 => {{1},{2},{3,6},{4,5}}
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => {{1,6},{2,5},{3},{4}}
 => {{1},{2,6},{3,4,5}}
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => {{1,2,6},{3},{4},{5}}
 => {{1},{2,3,4,5},{6}}
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => {{1,2,5,6},{3},{4}}
 => {{1},{2},{3,4,5},{6}}
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => {{1,6},{2,4,5},{3}}
 => {{1},{2,6},{3},{4,5}}
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => {{1,2,6},{3},{4,5}}
 => {{1},{2,4,5},{3},{6}}
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => {{1,2,4,6},{3},{5}}
 => {{1},{2,3},{4,5},{6}}
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => {{1,2,4,5,6},{3}}
 => {{1},{2},{3},{4,5},{6}}
 => 6
Description
The number of blocks ([[St000105]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000459
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2,1] => [2]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 3
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 3
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 4
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 5
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,2,1]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,2,1]
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => [2,1,1,1,1]
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => [2,2,1,1]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,5,6,1,4] => [2,1,1,1,1]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => [2,1,1,1,1]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,6,3] => [2,2,1,1]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,1,6,3,5] => [2,2,1,1]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,5,1,6,3] => [2,2,1,1]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,5,6,1,3] => [2,1,1,1,1]
 => 6
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,1,3,5] => [2,1,1,1,1]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,1,3,6,4] => [2,2,1,1]
 => 5
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,1,6,3,4] => [2,2,1,1]
 => 5
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,1,3,4] => [2,1,1,1,1]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5] => [2,1,1,1,1]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [2,2,1,1]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,4,6,2,5] => [2,2,1,1]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,1,5,2,6,4] => [2,2,2]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,5,6,2,4] => [2,2,1,1]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,1,6,2,4,5] => [2,2,1,1]
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,1,5,6,2] => [2,2,1,1]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,4,1,6,2,5] => [2,2,1,1]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => [2,2,1,1]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,1,2] => [2,1,1,1,1]
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5] => [2,1,1,1,1]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,5,1,2,6,4] => [2,2,1,1]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,5,1,6,2,4] => [2,2,1,1]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,5,6,1,2,4] => [2,1,1,1,1]
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,6,1,2,4,5] => [2,1,1,1,1]
 => 6
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St001492
(load all 4 compositions to match this statistic)
Mapping
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[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]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 5
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,1,1,0,0,0,0]
 => [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]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 6
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 5
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 5
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 6
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St000010
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2,1] => [2]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 3 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => [2,2,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,5,6,1,4] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,6,3] => [2,2,1,1]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,1,6,3,5] => [2,2,1,1]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,5,1,6,3] => [2,2,1,1]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,5,6,1,3] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,1,3,5] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,1,3,6,4] => [2,2,1,1]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,1,6,3,4] => [2,2,1,1]
 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,1,3,4] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [2,2,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,4,6,2,5] => [2,2,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,1,5,2,6,4] => [2,2,2]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,5,6,2,4] => [2,2,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,1,6,2,4,5] => [2,2,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,1,5,6,2] => [2,2,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,4,1,6,2,5] => [2,2,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => [2,2,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,1,2] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,5,1,2,6,4] => [2,2,1,1]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,5,1,6,2,4] => [2,2,1,1]
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,5,6,1,2,4] => [2,1,1,1,1]
 => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,6,1,2,4,5] => [2,1,1,1,1]
 => 5 = 6 - 1
Description
The length of the partition.
Matching statistic: St000093
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => [2,1] => [2] => ([],2)
 => 2 = 3 - 1
[1,1,0,0]
 => [1,2] => [2] => ([],2)
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [3] => ([],3)
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [3] => ([],3)
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,2] => ([(1,2)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [3] => ([],3)
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [3] => ([],3)
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4] => ([],4)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [4] => ([],4)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4] => ([],4)
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [4] => ([],4)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,3] => ([(2,3)],4)
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3] => ([(2,3)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4] => ([],4)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [4] => ([],4)
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,3] => ([(2,3)],4)
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4] => ([],4)
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4] => ([],4)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,4] => ([(3,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4] => ([(3,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5] => ([],5)
 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [5] => ([],5)
 => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [5] => ([],5)
 => 5 = 6 - 1
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000393
(load all 3 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [2,1] => 1 => 1 = 2 - 1
[1,0,1,0]
 => [3,1,2] => [3,2,1] => 11 => 2 = 3 - 1
[1,1,0,0]
 => [2,3,1] => [3,1,2] => 10 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,3,2,1] => 111 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,1,3] => 110 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,3,1,2] => 101 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,3,1] => 110 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 100 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,3,2,1,4] => 1110 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,4,2,1,3] => 1101 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [5,3,4,2,1] => 1110 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,2,1,3,4] => 1100 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,4,3,1,2] => 1011 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,3,1,2,4] => 1010 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [5,4,2,3,1] => 1101 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,2,5,3,1] => 1110 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,3,1,4] => 1100 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,4,1,2,3] => 1001 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,3,4,1,2] => 1010 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,4,1] => 1100 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,5,4,3,2,1] => 11111 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,4,3,2,1,5] => 11110 => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,5,3,2,1,4] => 11101 => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [6,4,5,3,2,1] => 11110 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,3,2,1,4,5] => 11100 => 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,5,4,2,1,3] => 11011 => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,4,2,1,3,5] => 11010 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [6,5,3,4,2,1] => 11101 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [5,3,6,4,2,1] => 11110 => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [6,3,4,2,1,5] => 11100 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,5,2,1,3,4] => 11001 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [6,4,5,2,1,3] => 11010 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [6,3,4,5,2,1] => 11100 => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,2,1,3,4,5] => 11000 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,5,4,3,1,2] => 10111 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6,4,3,1,2,5] => 10110 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6,5,3,1,2,4] => 10101 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [6,4,5,3,1,2] => 10110 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,3,1,2,4,5] => 10100 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [6,5,4,2,3,1] => 11011 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [6,4,2,3,1,5] => 11010 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2,6,5,3,1] => 11101 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [5,3,1,6,4,2] => 11110 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2,6,3,1,5] => 11100 => 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [6,5,2,3,1,4] => 11001 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [6,4,5,2,3,1] => 11010 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,2,4,6,3,1] => 11100 => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,3,1,4,5] => 11000 => 5 = 6 - 1
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000507
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2,1] => [[1],[2]]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [[1,2],[3]]
 => 2 = 3 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => [[1,3],[2]]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [[1,2],[3,4]]
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [[1,3],[2,4]]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [[1,3,4],[2]]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[1,2,3],[4,5]]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [[1,2,3],[4,5]]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [[1,3,4,5],[2]]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [[1,2,3,4,5],[6]]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => [[1,2,3,4],[5,6]]
 => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => [[1,2,3,5],[4,6]]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,5,6,1,4] => [[1,2,3,4],[5,6]]
 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => [[1,2,3,6],[4,5]]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,6,3] => [[1,2,4,5],[3,6]]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,1,6,3,5] => [[1,2,4],[3,5,6]]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,5,1,6,3] => [[1,2,3,5],[4,6]]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,5,6,1,3] => [[1,2,3,4],[5,6]]
 => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,1,3,5] => [[1,2,3],[4,5,6]]
 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,1,3,6,4] => [[1,2,5],[3,4,6]]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,1,6,3,4] => [[1,2,4],[3,5,6]]
 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,1,3,4] => [[1,2,3],[4,5,6]]
 => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5] => [[1,2,5,6],[3,4]]
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [[1,3,4,5],[2,6]]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,4,6,2,5] => [[1,3,4],[2,5,6]]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,1,5,2,6,4] => [[1,3,5],[2,4,6]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,5,6,2,4] => [[1,3,4],[2,5,6]]
 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,1,6,2,4,5] => [[1,3,5,6],[2,4]]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,1,5,6,2] => [[1,2,4,5],[3,6]]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,4,1,6,2,5] => [[1,2,4],[3,5,6]]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => [[1,2,3,5],[4,6]]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,1,2] => [[1,2,3,4],[5,6]]
 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5] => [[1,2,3],[4,5,6]]
 => 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,5,1,2,6,4] => [[1,2,5],[3,4,6]]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,5,1,6,2,4] => [[1,2,4],[3,5,6]]
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,5,6,1,2,4] => [[1,2,3],[4,5,6]]
 => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,6,1,2,4,5] => [[1,2,5,6],[3,4]]
 => 5 = 6 - 1
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000786
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => [2,1] => [2] => ([],2)
 => 2 = 3 - 1
[1,1,0,0]
 => [1,2] => [2] => ([],2)
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [3] => ([],3)
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [3] => ([],3)
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,2] => ([(1,2)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [3] => ([],3)
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [3] => ([],3)
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4] => ([],4)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [4] => ([],4)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4] => ([],4)
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [4] => ([],4)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,3] => ([(2,3)],4)
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3] => ([(2,3)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4] => ([],4)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [4] => ([],4)
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,3] => ([(2,3)],4)
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4] => ([],4)
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4] => ([],4)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,4] => ([(3,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4] => ([(3,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5] => ([],5)
 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [5] => ([],5)
 => 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [5] => ([],5)
 => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [5] => ([],5)
 => 5 = 6 - 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000863The length of the first row of the shifted shape of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001286The annihilation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000552The number of cut vertices of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001692The number of vertices with higher degree than the average degree in a graph. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000619The number of cyclic descents of a permutation. St000991The number of right-to-left minima of a permutation. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St001875The number of simple modules with projective dimension at most 1. St000245The number of ascents of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001820The size of the image of the pop stack sorting operator. St001298The number of repeated entries in the Lehmer code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001626The number of maximal proper sublattices of a lattice. St000907The number of maximal antichains of minimal length in a poset. St000213The number of weak exceedances (also weak excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001649The length of a longest trail in a graph. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St001668The number of points of the poset minus the width of the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000144The pyramid weight of the Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.