Processing math: 100%

Your data matches 141 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000528
(load all 2 compositions to match this statistic)
Mapping
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 2
([],3)
 => 1
([(1,2)],3)
 => 2
([(0,1),(0,2)],3)
 => 2
([(0,2),(2,1)],3)
 => 3
([(0,2),(1,2)],3)
 => 2
([],4)
 => 1
([(2,3)],4)
 => 2
([(1,2),(1,3)],4)
 => 2
([(0,1),(0,2),(0,3)],4)
 => 2
([(0,2),(0,3),(3,1)],4)
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
([(1,2),(2,3)],4)
 => 3
([(0,3),(3,1),(3,2)],4)
 => 3
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(3,2)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => 3
([],5)
 => 1
([(3,4)],5)
 => 2
([(2,3),(2,4)],5)
 => 2
([(1,2),(1,3),(1,4)],5)
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(4,2)],5)
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(2,3),(3,4)],5)
 => 3
([(1,4),(4,2),(4,3)],5)
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
([(2,4),(3,4)],5)
 => 2
([(1,4),(2,4),(4,3)],5)
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,3)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
Description
The height of a poset. This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 1
([],2)
 => [1,1]
 => 1
([(0,1)],2)
 => [2]
 => 2
([],3)
 => [1,1,1]
 => 1
([(1,2)],3)
 => [2,1]
 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => 2
([(0,2),(2,1)],3)
 => [3]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => 2
([],4)
 => [1,1,1,1]
 => 1
([(2,3)],4)
 => [2,1,1]
 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 3
([],5)
 => [1,1,1,1,1]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => 2
([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)
 => ?
 => ? = 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ? = 2
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
 => ?
 => ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ?
 => ? = 2
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ? = 2
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
([],8)
 => ?
 => ? = 1
([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 4
([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
 => ?
 => ? = 4
([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
 => ?
 => ? = 4
([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
 => ?
 => ? = 4
([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 4
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 3
([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
([(0,7),(1,7),(2,6),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(5,7)],8)
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
 => ?
 => ? = 3
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 3
([(0,3),(0,5),(1,2),(1,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6)],8)
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
 => ?
 => ? = 2
([(4,7),(5,6)],8)
 => ?
 => ? = 2
([(0,7),(1,6),(2,5),(3,4)],8)
 => ?
 => ? = 2
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => 1
([],2)
 => [1,1]
 => [2]
 => 1
([(0,1)],2)
 => [2]
 => [1,1]
 => 2
([],3)
 => [1,1,1]
 => [3]
 => 1
([(1,2)],3)
 => [2,1]
 => [2,1]
 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [2,1]
 => 2
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [2,1]
 => 2
([],4)
 => [1,1,1,1]
 => [4]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
([],5)
 => [1,1,1,1,1]
 => [5]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [3,2]
 => 2
([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ? = 2
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
 => ?
 => ?
 => ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ?
 => ?
 => ? = 2
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ? = 2
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 2
([],8)
 => ?
 => ?
 => ? = 1
([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
 => ?
 => ?
 => ? = 4
([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 2
([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 2
([(0,7),(1,7),(2,6),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
 => ?
 => ?
 => ? = 3
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 3
([(0,3),(0,5),(1,2),(1,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6)],8)
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
 => ?
 => ?
 => ? = 2
([(4,7),(5,6)],8)
 => ?
 => ?
 => ? = 2
([(0,7),(1,6),(2,5),(3,4)],8)
 => ?
 => ?
 => ? = 2
Description
The length of the partition.
Matching statistic: St000476
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([],2)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([],3)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
([(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
([],4)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 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),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([],5)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([],8)
 => ?
 => ?
 => ?
 => ? = 1
([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,7),(1,7),(2,6),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,5),(1,2),(1,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(4,7),(5,6)],8)
 => ?
 => ?
 => ?
 => ? = 2
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path. For each valley v in a Dyck path D there is a corresponding tunnel, which is the factor Tv=si…sj of D where si is the step after the first intersection of D with the line y=ht(v) to the left of sj. This statistic is ∑v(jv−iv)/2.
Matching statistic: St000378
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => [1]
 => 1
([],2)
 => [1,1]
 => [2]
 => [1,1]
 => 1
([(0,1)],2)
 => [2]
 => [1,1]
 => [2]
 => 2
([],3)
 => [1,1,1]
 => [3]
 => [1,1,1]
 => 1
([(1,2)],3)
 => [2,1]
 => [2,1]
 => [3]
 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [2,1]
 => [3]
 => 2
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => [2,1]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [2,1]
 => [3]
 => 2
([],4)
 => [1,1,1,1]
 => [4]
 => [1,1,1,1]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => [4]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => [4]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => [4]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => [3,1]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
([],5)
 => [1,1,1,1,1]
 => [5]
 => [1,1,1,1,1]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => [3,1,1]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [2,2,1]
 => [2,2,1]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [2,2,1]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [2,2,1]
 => [2,2,1]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [2,2,1]
 => [2,2,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => [5]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [3,2]
 => [5]
 => 2
([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([],8)
 => ?
 => ?
 => ?
 => ? = 1
([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,7),(1,7),(2,9),(3,10),(4,11),(5,12),(6,8),(7,12),(9,11),(10,9),(11,8),(12,10)],13)
 => [7,1,1,1,1,1,1]
 => [7,1,1,1,1,1,1]
 => [4,3,2,1,1,1,1]
 => ? = 7
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
 => [5,2,2,1,1,1,1]
 => [7,3,1,1,1]
 => [4,4,1,1,1,1,1]
 => ? = 5
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => [5,2,2,1,1,1,1]
 => [7,3,1,1,1]
 => [4,4,1,1,1,1,1]
 => ? = 5
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
 => [5,2,2,1,1,1,1]
 => [7,3,1,1,1]
 => [4,4,1,1,1,1,1]
 => ? = 5
([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
 => [4,3,2,1,1,1,1]
 => [7,3,2,1]
 => [6,2,1,1,1,1,1]
 => ? = 4
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,7),(1,7),(2,6),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,3),(0,5),(1,2),(1,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells c in the diagram of an integer partition λ for which arm(c)−leg(c)∈{0,1}. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000288
(load all 2 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => 10 => 1
([],2)
 => [1,1]
 => [2]
 => 100 => 1
([(0,1)],2)
 => [2]
 => [1,1]
 => 110 => 2
([],3)
 => [1,1,1]
 => [3]
 => 1000 => 1
([(1,2)],3)
 => [2,1]
 => [2,1]
 => 1010 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [2,1]
 => 1010 => 2
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 1110 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [2,1]
 => 1010 => 2
([],4)
 => [1,1,1,1]
 => [4]
 => 10000 => 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 10010 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [3,1]
 => 10010 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [3,1]
 => 10010 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => 10110 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 10110 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 10110 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 10110 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 10010 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 10110 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 10010 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 1100 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 1100 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 1100 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 11110 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 10110 => 3
([],5)
 => [1,1,1,1,1]
 => [5]
 => 100000 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 100010 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 100010 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 100010 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 100010 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => 101110 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [2,2,1]
 => 11010 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 11010 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [2,2,1]
 => 11010 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 100010 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [2,2,1]
 => 11010 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 100010 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 100010 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 10100 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [3,2]
 => 10100 => 2
([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
 => ?
 => ?
 => ? => ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 2
([],8)
 => ?
 => ?
 => ? => ? = 1
([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => [6,1,1,1,1,1]
 => [6,1,1,1,1,1]
 => 100000111110 => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => [4,2,2,1,1,1]
 => [6,3,1,1]
 => 1000100110 => ? = 4
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => [4,3,1,1,1,1]
 => [6,2,2,1]
 => 1000011010 => ? = 4
([(0,7),(1,7),(2,9),(3,10),(4,11),(5,12),(6,8),(7,12),(9,11),(10,9),(11,8),(12,10)],13)
 => [7,1,1,1,1,1,1]
 => [7,1,1,1,1,1,1]
 => 10000001111110 => ? = 7
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [7,2,1,1,1,1]
 => 1000001011110 => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [7,2,1,1,1,1]
 => 1000001011110 => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
 => [6,2,1,1,1,1,1]
 => [7,2,1,1,1,1]
 => 1000001011110 => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
 => [5,2,2,1,1,1,1]
 => [7,3,1,1,1]
 => 100001001110 => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
 => [5,3,1,1,1,1,1]
 => [7,2,2,1,1]
 => 100000110110 => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
 => [6,2,1,1,1,1,1]
 => [7,2,1,1,1,1]
 => 1000001011110 => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => [5,2,2,1,1,1,1]
 => [7,3,1,1,1]
 => 100001001110 => ? = 5
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
 => [5,2,2,1,1,1,1]
 => [7,3,1,1,1]
 => 100001001110 => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
 => [5,3,1,1,1,1,1]
 => [7,2,2,1,1]
 => 100000110110 => ? = 5
([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
 => [4,3,2,1,1,1,1]
 => [7,3,2,1]
 => 10000101010 => ? = 4
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 2
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000734
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1]]
 => 1
([],2)
 => [1,1]
 => [[1],[2]]
 => 1
([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => 2
([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => 2
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ? = 2
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
 => ?
 => ?
 => ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ?
 => ?
 => ? = 2
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ? = 2
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 2
([],8)
 => ?
 => ?
 => ? = 1
([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
 => ?
 => ?
 => ? = 4
([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 4
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => [6,1,1,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
 => ? = 6
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => [5,2,1,1,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
 => ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => [5,2,1,1,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
 => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => [5,2,1,1,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
 => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => [4,2,2,1,1,1]
 => [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
 => ? = 4
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => [4,3,1,1,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
 => ? = 4
([(0,7),(1,7),(2,9),(3,10),(4,11),(5,12),(6,8),(7,12),(9,11),(10,9),(11,8),(12,10)],13)
 => [7,1,1,1,1,1,1]
 => [[1,2,3,4,5,6,7],[8],[9],[10],[11],[12],[13]]
 => ? = 7
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10],[11],[12],[13]]
 => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10],[11],[12],[13]]
 => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
 => [6,2,1,1,1,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10],[11],[12],[13]]
 => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
 => [5,2,2,1,1,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
 => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
 => [5,3,1,1,1,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13]]
 => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
 => [6,2,1,1,1,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10],[11],[12],[13]]
 => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => [5,2,2,1,1,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
 => ? = 5
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
 => [5,2,2,1,1,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
 => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
 => [5,3,1,1,1,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13]]
 => ? = 5
([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
 => [4,3,2,1,1,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12],[13]]
 => ? = 4
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 3
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000013
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([],2)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([],3)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
([],4)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([],5)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([],8)
 => ?
 => ?
 => ?
 => ? = 1
([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => [6,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => [5,2,1,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => [5,2,1,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => [5,2,1,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => [4,2,2,1,1,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 4
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => [4,3,1,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
([(0,7),(1,7),(2,9),(3,10),(4,11),(5,12),(6,8),(7,12),(9,11),(10,9),(11,8),(12,10)],13)
 => [7,1,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 7
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
 => [6,2,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
 => [5,2,2,1,1,1,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
 => [5,3,1,1,1,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
 => [6,2,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => [5,2,2,1,1,1,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 5
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
 => [5,2,2,1,1,1,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
 => [5,3,1,1,1,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 5
([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
 => [4,3,2,1,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 4
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
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: St000141
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
([],2)
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
([],3)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
([(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
([],4)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
([],5)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
 => ?
 => ?
 => ? => ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ? => ? = 2
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 2
([],8)
 => ?
 => ?
 => ? => ? = 1
([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 4
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => [6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => ? = 6
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [6,3,2,4,5,7,1] => ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [6,3,2,4,5,7,1] => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [6,3,2,4,5,7,1] => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => [4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [5,3,4,2,6,7,1] => ? = 4
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => [4,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,6,7,1] => ? = 4
([(0,7),(1,7),(2,9),(3,10),(4,11),(5,12),(6,8),(7,12),(9,11),(10,9),(11,8),(12,10)],13)
 => [7,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,6,7,1] => ? = 7
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [7,3,2,4,5,6,8,1] => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [7,3,2,4,5,6,8,1] => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
 => [6,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [7,3,2,4,5,6,8,1] => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [6,3,4,2,5,7,8,1] => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
 => [5,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [6,4,2,3,5,7,8,1] => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
 => [6,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [7,3,2,4,5,6,8,1] => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [6,3,4,2,5,7,8,1] => ? = 5
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [6,3,4,2,5,7,8,1] => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
 => [5,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [6,4,2,3,5,7,8,1] => ? = 5
([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
 => [4,3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [5,4,3,2,6,7,8,1] => ? = 4
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? => ? = 3
Description
The maximum drop size of a permutation. The maximum drop size of a permutation π of [n]={1,2,…,n} is defined to be the maximum value of i−π(i).
Matching statistic: St000733
(load all 2 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1]]
 => [[1]]
 => 1
([],2)
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
([(0,1)],2)
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
([(1,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 2
([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,6),(0,7),(1,6),(1,7),(6,2),(6,3),(6,4),(6,5),(7,2),(7,3),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([],8)
 => ?
 => ?
 => ?
 => ? = 1
([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 4
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => [6,1,1,1,1,1]
 => [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
 => ?
 => ? = 6
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => [5,2,1,1,1,1]
 => [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6,8],[9],[10],[11]]
 => ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => [5,2,1,1,1,1]
 => [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6,8],[9],[10],[11]]
 => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => [5,2,1,1,1,1]
 => [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6,8],[9],[10],[11]]
 => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => [4,2,2,1,1,1]
 => [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
 => ?
 => ? = 4
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => [4,3,1,1,1,1]
 => [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
 => ?
 => ? = 4
([(0,7),(1,7),(2,9),(3,10),(4,11),(5,12),(6,8),(7,12),(9,11),(10,9),(11,8),(12,10)],13)
 => [7,1,1,1,1,1,1]
 => [[1,8,9,10,11,12,13],[2],[3],[4],[5],[6],[7]]
 => ?
 => ? = 7
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [[1,7,10,11,12,13],[2,9],[3],[4],[5],[6],[8]]
 => ?
 => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
 => [6,2,1,1,1,1,1]
 => [[1,7,10,11,12,13],[2,9],[3],[4],[5],[6],[8]]
 => ?
 => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
 => [6,2,1,1,1,1,1]
 => [[1,7,10,11,12,13],[2,9],[3],[4],[5],[6],[8]]
 => ?
 => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
 => [5,2,2,1,1,1,1]
 => [[1,6,11,12,13],[2,8],[3,10],[4],[5],[7],[9]]
 => ?
 => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
 => [5,3,1,1,1,1,1]
 => [[1,7,8,12,13],[2,10,11],[3],[4],[5],[6],[9]]
 => ?
 => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
 => [6,2,1,1,1,1,1]
 => [[1,7,10,11,12,13],[2,9],[3],[4],[5],[6],[8]]
 => ?
 => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => [5,2,2,1,1,1,1]
 => [[1,6,11,12,13],[2,8],[3,10],[4],[5],[7],[9]]
 => ?
 => ? = 5
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
 => [5,2,2,1,1,1,1]
 => [[1,6,11,12,13],[2,8],[3,10],[4],[5],[7],[9]]
 => ?
 => ? = 5
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
 => [5,3,1,1,1,1,1]
 => [[1,7,8,12,13],[2,10,11],[3],[4],[5],[6],[9]]
 => ?
 => ? = 5
([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
 => [4,3,2,1,1,1,1]
 => [[1,6,9,13],[2,8,12],[3,11],[4],[5],[7],[10]]
 => ?
 => ? = 4
([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
Description
The row containing the largest entry of a standard tableau.
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000054The first entry of the permutation. St000157The number of descents of a standard tableau. St000676The number of odd rises of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000691The number of changes of a binary word. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000738The first entry in the last row of a standard tableau. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000808The number of up steps of the associated bargraph. St000025The number of initial rises of a Dyck path. St000383The last part of an integer composition. St000505The biggest entry in the block containing the 1. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000971The smallest closer of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000439The position of the first down step of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001777The number of weak descents in an integer composition. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000503The maximal difference between two elements in a common block. St000874The position of the last double rise in a Dyck path. St000702The number of weak deficiencies of a permutation. St001029The size of the core of a graph. St000653The last descent of a permutation. St001497The position of the largest weak excedence of 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. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000031The number of cycles in the cycle decomposition of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000083The number of left oriented leafs of a binary tree except the first one. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000740The last entry of a permutation. St000741The Colin de Verdière graph invariant. St000093The cardinality of a maximal independent set of vertices of a graph. St000746The number of pairs with odd minimum in a perfect matching. St000308The height of the tree associated to a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000080The rank of the poset. St000062The length of the longest increasing subsequence of the permutation. St000087The number of induced subgraphs. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000926The clique-coclique number of a graph. St000991The number of right-to-left minima of a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001812The biclique partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001323The independence gap of a graph. St001651The Frankl number of a lattice. St001720The minimal length of a chain of small intervals in a lattice.