Your data matches 54 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000381
(load all 32 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => 1
([],2)
 => [2] => 2
([(0,1)],2)
 => [1,1] => 1
([],3)
 => [3] => 3
([(1,2)],3)
 => [1,2] => 2
([(0,2),(1,2)],3)
 => [1,1,1] => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => 2
([],4)
 => [4] => 4
([(2,3)],4)
 => [1,3] => 3
([(1,3),(2,3)],4)
 => [1,1,2] => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => 2
([(0,3),(1,2)],4)
 => [2,2] => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 3
([],5)
 => [5] => 5
([(3,4)],5)
 => [1,4] => 4
([(2,4),(3,4)],5)
 => [1,1,3] => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => 3
([(1,4),(2,3)],5)
 => [2,3] => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1]
 => 1
([],2)
 => [2] => [2]
 => 2
([(0,1)],2)
 => [1,1] => [1,1]
 => 1
([],3)
 => [3] => [3]
 => 3
([(1,2)],3)
 => [1,2] => [2,1]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => 2
([],4)
 => [4] => [4]
 => 4
([(2,3)],4)
 => [1,3] => [3,1]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,1,1]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [2,2]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,1,1]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => 3
([],5)
 => [5] => [5]
 => 5
([(3,4)],5)
 => [1,4] => [4,1]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [3,1,1]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [3,2]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 2
Description
The largest part of an integer partition.
Matching statistic: St000010
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1]
 => [1]
 => 1
([],2)
 => [2] => [2]
 => [1,1]
 => 2
([(0,1)],2)
 => [1,1] => [1,1]
 => [2]
 => 1
([],3)
 => [3] => [3]
 => [1,1,1]
 => 3
([(1,2)],3)
 => [1,2] => [2,1]
 => [2,1]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => [3]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => [2,1]
 => 2
([],4)
 => [4] => [4]
 => [1,1,1,1]
 => 4
([(2,3)],4)
 => [1,3] => [3,1]
 => [2,1,1]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1,1]
 => [3,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,1,1]
 => [3,1]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [4]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,1,1]
 => [3,1]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [3,1]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => [2,1,1]
 => 3
([],5)
 => [5] => [5]
 => [1,1,1,1,1]
 => 5
([(3,4)],5)
 => [1,4] => [4,1]
 => [2,1,1,1]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1]
 => [3,1,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [3,1,1]
 => [3,1,1]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [3,2]
 => [2,2,1]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2]
 => [2,2,1]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => [2,2,1]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 2
Description
The length of the partition.
Matching statistic: St000013
(load all 9 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1,0]
 => [1,0]
 => 1
([],2)
 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 2
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
([(1,4),(2,3),(3,4)],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]
 => 2
([(0,1),(2,4),(3,4)],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]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [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
([(1,4),(2,3),(2,4),(3,4)],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]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [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
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [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
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [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
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [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),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [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
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [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
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [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
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [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
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000676
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1]
 => [1,0]
 => 1
([],2)
 => [2] => [2]
 => [1,0,1,0]
 => 2
([(0,1)],2)
 => [1,1] => [1,1]
 => [1,1,0,0]
 => 1
([],3)
 => [3] => [3]
 => [1,0,1,0,1,0]
 => 3
([(1,2)],3)
 => [1,2] => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 2
([],4)
 => [4] => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(2,3)],4)
 => [1,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([],5)
 => [5] => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(3,4)],5)
 => [1,4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000684
(load all 2 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1] => [1,0]
 => 1
([],2)
 => [2] => [1,1] => [1,0,1,0]
 => 2
([(0,1)],2)
 => [1,1] => [2] => [1,1,0,0]
 => 1
([],3)
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3
([(1,2)],3)
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 2
([],4)
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([(2,3)],4)
 => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
([],5)
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(3,4)],5)
 => [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
Description
The global dimension of the LNakayama algebra associated to a Dyck path. An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$. The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$. One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0]. Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$. Examples: * For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192. * For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
Matching statistic: St000734
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1]
 => [[1]]
 => 1
([],2)
 => [2] => [2]
 => [[1,2]]
 => 2
([(0,1)],2)
 => [1,1] => [1,1]
 => [[1],[2]]
 => 1
([],3)
 => [3] => [3]
 => [[1,2,3]]
 => 3
([(1,2)],3)
 => [1,2] => [2,1]
 => [[1,2],[3]]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => [[1,2],[3]]
 => 2
([],4)
 => [4] => [4]
 => [[1,2,3,4]]
 => 4
([(2,3)],4)
 => [1,3] => [3,1]
 => [[1,2,3],[4]]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => [[1,2,3],[4]]
 => 3
([],5)
 => [5] => [5]
 => [[1,2,3,4,5]]
 => 5
([(3,4)],5)
 => [1,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000983
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => 1 => 1 => 1
([],2)
 => [2] => 10 => 01 => 2
([(0,1)],2)
 => [1,1] => 11 => 11 => 1
([],3)
 => [3] => 100 => 101 => 3
([(1,2)],3)
 => [1,2] => 110 => 011 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => 111 => 111 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => 101 => 001 => 2
([],4)
 => [4] => 1000 => 0101 => 4
([(2,3)],4)
 => [1,3] => 1100 => 1011 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => 1110 => 0111 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => 1101 => 0011 => 2
([(0,3),(1,2)],4)
 => [2,2] => 1010 => 1001 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => 1111 => 1111 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => 1010 => 1001 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1111 => 1111 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => 1101 => 0011 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 0001 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 1001 => 1101 => 3
([],5)
 => [5] => 10000 => 10101 => 5
([(3,4)],5)
 => [1,4] => 11000 => 01011 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => 11100 => 10111 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => 11010 => 10011 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => 11001 => 11011 => 3
([(1,4),(2,3)],5)
 => [2,3] => 10100 => 01001 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => 11110 => 01111 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => 11110 => 01111 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => 10100 => 01001 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 11110 => 01111 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => 11101 => 00111 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => 11010 => 10011 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 10110 => 10001 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => 11101 => 00111 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11001 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 10110 => 10001 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => 11011 => 00011 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => 10101 => 11001 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => 10010 => 01101 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => 11011 => 00011 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 00001 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11001 => 2
Description
The length of the longest alternating subword. This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => 1 => 0 => 0 = 1 - 1
([],2)
 => [2] => 10 => 01 => 1 = 2 - 1
([(0,1)],2)
 => [1,1] => 11 => 00 => 0 = 1 - 1
([],3)
 => [3] => 100 => 011 => 2 = 3 - 1
([(1,2)],3)
 => [1,2] => 110 => 001 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [1,1,1] => 111 => 000 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => 101 => 010 => 1 = 2 - 1
([],4)
 => [4] => 1000 => 0111 => 3 = 4 - 1
([(2,3)],4)
 => [1,3] => 1100 => 0011 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => [1,1,2] => 1110 => 0001 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => 1101 => 0010 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => 1010 => 0101 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => 1010 => 0101 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => 1101 => 0010 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 0100 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 1001 => 0110 => 2 = 3 - 1
([],5)
 => [5] => 10000 => 01111 => 4 = 5 - 1
([(3,4)],5)
 => [1,4] => 11000 => 00111 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [1,1,3] => 11100 => 00011 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => 11010 => 00101 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => 11001 => 00110 => 2 = 3 - 1
([(1,4),(2,3)],5)
 => [2,3] => 10100 => 01011 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => 10100 => 01011 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => 11010 => 00101 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 10110 => 01001 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 10110 => 01001 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => 10101 => 01010 => 1 = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => 10010 => 01101 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 1 = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001039
(load all 2 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1]
 => [1,0]
 => ? = 1
([],2)
 => [2] => [2]
 => [1,0,1,0]
 => 2
([(0,1)],2)
 => [1,1] => [1,1]
 => [1,1,0,0]
 => 1
([],3)
 => [3] => [3]
 => [1,0,1,0,1,0]
 => 3
([(1,2)],3)
 => [1,2] => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 2
([],4)
 => [4] => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(2,3)],4)
 => [1,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([],5)
 => [5] => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(3,4)],5)
 => [1,4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000439The position of the first down step of a Dyck path. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000094The depth of an ordered tree. St000025The number of initial rises of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001809The index of the step at the first peak of maximal height in a Dyck path. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000662The staircase size of the code of a permutation. St000306The bounce count of a Dyck path. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St000141The maximum drop size of a permutation. St000028The number of stack-sorts needed to sort a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000308The height of the tree associated to a permutation. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001652The length of a longest interval of consecutive numbers. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001589The nesting number of a perfect matching.