- FindStat

Your data matches 35 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000643
(load all 2 compositions to match this statistic)

Mapping

St000643: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],2)
 => 2
([(0,1)],2)
 => 3
([],3)
 => 2
([(1,2)],3)
 => 6
([(0,1),(0,2)],3)
 => 3
([(0,2),(2,1)],3)
 => 4
([(0,2),(1,2)],3)
 => 3
([],4)
 => 2
([(2,3)],4)
 => 6
([(1,2),(1,3)],4)
 => 6
([(0,1),(0,2),(0,3)],4)
 => 3
([(0,2),(0,3),(3,1)],4)
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(1,2),(2,3)],4)
 => 4
([(0,3),(3,1),(3,2)],4)
 => 4
([(1,3),(2,3)],4)
 => 6
([(0,3),(1,3),(3,2)],4)
 => 4
([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => 3
([(0,3),(1,2),(1,3)],4)
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
([(0,3),(2,1),(3,2)],4)
 => 5
([(0,3),(1,2),(2,3)],4)
 => 7
([],5)
 => 2
([(3,4)],5)
 => 6
([(2,3),(2,4)],5)
 => 6
([(1,2),(1,3),(1,4)],5)
 => 6
([(0,1),(0,2),(0,3),(0,4)],5)
 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(4,2)],5)
 => 14
([(0,3),(0,4),(4,1),(4,2)],5)
 => 7
([(1,2),(1,3),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(2,3),(3,4)],5)
 => 4
([(1,4),(4,2),(4,3)],5)
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => 4
([(2,4),(3,4)],5)
 => 6
([(1,4),(2,4),(4,3)],5)
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => 4
([(1,4),(2,4),(3,4)],5)
 => 6
([(0,4),(1,4),(2,4),(4,3)],5)
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3)],5)
 => 6
([(0,4),(1,3),(2,3),(2,4)],5)
 => 8
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 5

Description

The size of the largest orbit of antichains under Panyushev complementation.

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 64%

Values

([],2)
 => [2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2 = 3 - 1
([],3)
 => [2,2,2,2]
 => [4,4]
 => [4]
 => 1 = 2 - 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 5 = 6 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [8,8]
 => [8]
 => 1 = 2 - 1
([(2,3)],4)
 => [6,6]
 => [2,2,2,2,2,2]
 => [2,2,2,2,2]
 => 5 = 6 - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => [3,1,1,1,1]
 => 5 = 6 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [4,1]
 => 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 6 = 7 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 3 = 4 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 3 = 4 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => [3,1,1,1,1]
 => 5 = 6 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 3 = 4 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [4,1]
 => 2 = 3 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3,3]
 => [3,3]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => [3,1]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 6 = 7 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [16,16]
 => [16]
 => 1 = 2 - 1
([(3,4)],5)
 => [6,6,6,6]
 => [4,4,4,4,4,4]
 => [4,4,4,4,4]
 => ? = 6 - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => 5 = 6 - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [7,7,1,1,1,1]
 => [7,1,1,1,1]
 => 5 = 6 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [8,8,1]
 => [8,1]
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => 6 = 7 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => [3,1,1,1,1,1]
 => 6 = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [4,1,1]
 => 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => 13 = 14 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => [3,1,1,1,1,1]
 => 6 = 7 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3,3,1]
 => [3,3,1]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [2,2,2,1]
 => 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [3,1,1]
 => 3 = 4 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4,4]
 => [4,4,4]
 => 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [4,1,1]
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [3,1,1]
 => 3 = 4 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [7,7,1,1,1,1]
 => [7,1,1,1,1]
 => 5 = 6 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [4,1,1]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [8,8,1]
 => [8,1]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [4,4,4,1,1,1]
 => [4,4,1,1,1]
 => 5 = 6 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [3,3,2,1,1,1,1,1]
 => [3,2,1,1,1,1,1]
 => 7 = 8 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [4,4,2,1,1]
 => [4,2,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [5,5,1]
 => [5,1]
 => 2 = 3 - 1
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 2 - 1
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ?
 => ? = 6 - 1
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [16,16,1]
 => ?
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => [4,4,4,4,4,4,1]
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => [6,6,2,2,2,2,1]
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => [7,7,1,1,1,1,1]
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => [8,8,1,1]
 => ?
 => ? = 4 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ?
 => ? = 14 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => [6,6,2,2,2,2,1]
 => ?
 => ? = 7 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 4 - 1
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ?
 => ? = 14 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => [7,7,1,1,1,1,1]
 => ?
 => ? = 7 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 4 - 1
([(1,4),(1,5),(4,3),(5,2)],6)
 => [6,6,4,4]
 => [4,4,4,4,2,2]
 => ?
 => ? = 6 - 1
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ?
 => ?
 => ? = 4 - 1
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 4 - 1
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 4 - 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => [8,8,1,1]
 => ?
 => ? = 4 - 1
([(2,3),(3,5),(5,4)],6)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2,2,2]
 => ? = 10 - 1
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 4 - 1
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,5),(2,5),(3,5),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => [8,8,1,1]
 => ?
 => ? = 4 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [16,16,1]
 => ?
 => ? = 3 - 1
([(0,5),(1,5),(2,5),(3,4)],6)
 => [6,6,6,3,3,3]
 => ?
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [7,2,2,2,2,2,2]
 => [7,7,1,1,1,1,1]
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
 => [6,6,6,5,3]
 => ?
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [7,6,6,6]
 => [4,4,4,4,4,4,1]
 => ?
 => ? = 7 - 1
([(1,5),(2,5),(3,4)],6)
 => [6,6,6,6,6]
 => [5,5,5,5,5,5]
 => [5,5,5,5,5]
 => ? = 6 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [8,8,6,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [10,6,2,2,2,2]
 => ?
 => ?
 => ? = 10 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,5),(2,4),(3,4),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
([(0,5),(1,5),(2,3),(5,4)],6)
 => [12,6]
 => ?
 => ?
 => ? = 12 - 1
([(1,5),(2,5),(3,4),(3,5)],6)
 => [10,6,6,6]
 => ?
 => ?
 => ? = 10 - 1
([(0,5),(1,5),(2,3),(2,5),(3,4)],6)
 => [8,4,4,3]
 => ?
 => ?
 => ? = 8 - 1
([(0,5),(1,5),(2,3),(2,4)],6)
 => [6,6,3,3,3,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
 => [6,6,5,3,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
 => [10,6,2,2,2]
 => ?
 => ?
 => ? = 10 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => [6,6,2,2,2,2,1]
 => ?
 => ? = 7 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [14,6,6]
 => ?
 => ?
 => ? = 14 - 1
([(0,5),(1,5),(2,3),(3,4)],6)
 => [12,4,4]
 => ?
 => ?
 => ? = 12 - 1

Description

The length of the partition.

Matching statistic: St000147

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 64%

Values

([],2)
 => [2,2]
 => 2
([(0,1)],2)
 => [3]
 => 3
([],3)
 => [2,2,2,2]
 => 2
([(1,2)],3)
 => [6]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => 3
([],4)
 => [2,2,2,2,2,2,2,2]
 => 2
([(2,3)],4)
 => [6,6]
 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => 6
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 4
([(1,2),(2,3)],4)
 => [4,4]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 4
([(1,3),(2,3)],4)
 => [6,2,2]
 => 6
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 3
([(0,3),(1,2)],4)
 => [3,3,3]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 7
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 2
([(3,4)],5)
 => [6,6,6,6]
 => ? = 6
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => ? = 6
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 6
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => ? = 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 4
([(1,3),(1,4),(4,2)],5)
 => [14]
 => 14
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => 7
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => 4
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 4
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => ? = 6
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 6
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => ? = 3
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => 6
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => 8
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => 5
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => 7
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => 6
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => 7
([(1,4),(2,3)],5)
 => [6,6,6]
 => ? = 6
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => 10
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => 5
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 2
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ? = 6
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ? = 6
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 6
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ? = 4
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ? = 14
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ? = 7
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ? = 14
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ? = 4
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => ? = 7
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ? = 14
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ? = 14
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ? = 7
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ? = 4
([(1,4),(1,5),(4,3),(5,2)],6)
 => [6,6,4,4]
 => ? = 6
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => [10,4,4]
 => ? = 10
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ? = 4
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ? = 4
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ? = 4
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ? = 4
([(2,3),(3,5),(5,4)],6)
 => [10,10]
 => ? = 10
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ? = 6
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ? = 4
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 6
([(1,5),(2,5),(3,5),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ? = 4
([(1,5),(2,5),(3,5),(4,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 6
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ? = 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 3
([(0,5),(1,5),(2,5),(3,4)],6)
 => [6,6,6,3,3,3]
 => ? = 6
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [7,2,2,2,2,2,2]
 => ? = 7
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
 => [6,6,6,5,3]
 => ? = 6
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [7,6,6,6]
 => ? = 7
([(1,5),(2,5),(3,4)],6)
 => [6,6,6,6,6]
 => ? = 6
([(1,5),(2,4),(3,4),(3,5)],6)
 => [8,8,6,2,2]
 => ? = 8
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [10,6,2,2,2,2]
 => ? = 10
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,2,2,2,2,2,2,2,2]
 => ? = 6

Description

The largest part of an integer partition.

Matching statistic: St000013

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 40%

Values

([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 2
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => 6
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => 6
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 6
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 4
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 14
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => 7
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 4
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 4
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 3
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 8
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 5
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => 7
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => 6
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 7
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 6
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0]
 => ? = 10
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 4
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 5
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 14
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 12
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 2
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ?
 => ? = 6
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 4
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ?
 => ? = 14
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 7
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 14
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [6,4,3,3]
 => [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ?
 => ? = 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,4,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
 => ? = 8
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 5
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 8
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 5
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [10,7]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ?
 => ? = 10
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ?
 => ? = 7
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ?
 => ? = 14
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 14
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 4
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => [10,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ?
 => ? = 10
([(1,4),(1,5),(4,3),(5,2)],6)
 => [6,6,4,4]
 => ?
 => ?
 => ? = 6
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => [10,4,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0]
 => ? = 10
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,4,2,2,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 4
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [6,4,3,3]
 => [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ?
 => ? = 6
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => [8,4,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
 => ? = 8
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 5
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ?
 => ?
 => ? = 4

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: St000474

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 44%

Values

([],2)
 => [2,2]
 => 2
([(0,1)],2)
 => [3]
 => 3
([],3)
 => [2,2,2,2]
 => 2
([(1,2)],3)
 => [6]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => 3
([],4)
 => [2,2,2,2,2,2,2,2]
 => ? = 2
([(2,3)],4)
 => [6,6]
 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => 6
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 4
([(1,2),(2,3)],4)
 => [4,4]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 4
([(1,3),(2,3)],4)
 => [6,2,2]
 => 6
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 3
([(0,3),(1,2)],4)
 => [3,3,3]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 7
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 2
([(3,4)],5)
 => [6,6,6,6]
 => ? = 6
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => ? = 6
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 6
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => ? = 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => ? = 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 4
([(1,3),(1,4),(4,2)],5)
 => [14]
 => ? = 14
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => 7
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => 4
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => ? = 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 4
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => ? = 6
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 6
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => ? = 3
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => ? = 6
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => ? = 8
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => 5
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => 7
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => ? = 6
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => ? = 7
([(1,4),(2,3)],5)
 => [6,6,6]
 => ? = 6
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => ? = 10
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => ? = 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => 4
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => 10
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => ? = 6
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => ? = 6
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => 5
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => 7
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => ? = 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [14]
 => ? = 14
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 2
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ? = 6
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ? = 6
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 6
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ? = 4
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ? = 14
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ? = 7
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ? = 14
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ? = 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [6,4,3,3]
 => ? = 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,4,2]
 => ? = 8
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,4,2,2]
 => ? = 5
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [6,5,4]
 => ? = 6
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => ? = 5
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => ? = 7
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [10,7]
 => ? = 10
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => ? = 7
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ? = 14
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ? = 14
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ? = 7
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ? = 4
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => [10,2,2]
 => ? = 10

Description

Dyson's crank of a partition. Let

$\lambda$ be a partition and let

$o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let

$\mu(\lambda)$ be the number of parts that are strictly larger than

$o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as

$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$

Matching statistic: St000288
(load all 2 compositions to match this statistic)

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 36%

Values

([],2)
 => [2,2]
 => [2,2]
 => 1100 => 2
([(0,1)],2)
 => [3]
 => [1,1,1]
 => 1110 => 3
([],3)
 => [2,2,2,2]
 => [4,4]
 => 110000 => 2
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => 1111110 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => 11010 => 3
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => 11110 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => 11010 => 3
([],4)
 => [2,2,2,2,2,2,2,2]
 => [8,8]
 => 1100000000 => ? = 2
([(2,3)],4)
 => [6,6]
 => [2,2,2,2,2,2]
 => 11111100 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => 110011110 => 6
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 1100010 => 3
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 11111110 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 4
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => 111100 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 4
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => 110011110 => 6
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 4
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 1100010 => 3
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3,3]
 => 111000 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => 1110110 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => 110010 => 3
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => 111110 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 11111110 => 7
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [16,16]
 => 110000000000000000 => ? = 2
([(3,4)],5)
 => [6,6,6,6]
 => [4,4,4,4,4,4]
 => 1111110000 => ? = 6
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [6,6,2,2,2,2]
 => 110000111100 => ? = 6
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [7,7,1,1,1,1]
 => 1100000011110 => ? = 6
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [8,8,1]
 => 11000000010 => ? = 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => 111111010 => 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => 1100111110 => ? = 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 11000110 => 4
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => 111111111111110 => ? = 14
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => 1100111110 => ? = 7
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => 11001100 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3,3,1]
 => 1110010 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 1111010 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 4
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4,4]
 => 11110000 => 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => 11001100 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 11000110 => 4
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [6,6,2,2,2,2]
 => 110000111100 => ? = 6
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => 11001100 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 4
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [7,7,1,1,1,1]
 => 1100000011110 => ? = 6
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 11000110 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [8,8,1]
 => 11000000010 => ? = 3
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [4,4,4,1,1,1]
 => 1110001110 => ? = 6
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [3,3,2,1,1,1,1,1]
 => 11010111110 => ? = 8
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [4,4,2,1,1]
 => 110010110 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [5,5,1]
 => 11000010 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 5
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => 1100111110 => ? = 7
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => 111011010 => ? = 6
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => 111111010 => 7
([(1,4),(2,3)],5)
 => [6,6,6]
 => [3,3,3,3,3,3]
 => 111111000 => 6
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [2,2,2,2,2,2,1,1,1,1]
 => 111111011110 => ? = 10
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [2,2,2,1,1,1,1,1]
 => 1110111110 => 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 1111010 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [5,5,1,1,1,1]
 => 11000011110 => ? = 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => 110111110 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 4
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => 11111111110 => 10
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [4,4,4,1,1,1]
 => 1110001110 => ? = 6
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => 111011010 => ? = 6
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 1111010 => 5
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [2,2,1,1,1,1,1,1,1,1]
 => 110111111110 => ? = 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => 1100111110 => ? = 7
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [3,3,2,1,1,1,1,1]
 => 11010111110 => ? = 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [4,4,2,1,1]
 => 110010110 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [5,5,1]
 => 11000010 => 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => 11111111110 => 10
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => 111111111111110 => ? = 14
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => 1111111111110 => ? = 12
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? => ? = 2
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ? => ? = 6
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? => ? = 6
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? => ? = 6
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [16,16,1]
 => ? => ? = 3
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => [4,4,4,4,4,4,1]
 => ? => ? = 7
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => [6,6,2,2,2,2,1]
 => ? => ? = 7
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => [7,7,1,1,1,1,1]
 => ? => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => [8,8,1,1]
 => ? => ? = 4
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ? => ? = 14
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => [6,6,2,2,2,2,1]
 => ? => ? = 7
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? => ? = 14
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? => ? = 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [6,4,3,3]
 => [4,4,4,2,1,1]
 => 1110010110 => ? = 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,4,2]
 => [3,3,2,2,1,1,1,1]
 => 11011011110 => ? = 8
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2,2,2,2]
 => [5,5,1,1]
 => 110000110 => ? = 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [8,2,2]
 => [3,3,1,1,1,1,1,1]
 => 11001111110 => ? = 8
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => [3,3,3,3,3,3,1]
 => 1111110010 => ? = 7
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [10,7]
 => [2,2,2,2,2,2,2,1,1,1]
 => 111111101110 => ? = 10
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => [5,5,1,1,1,1,1]
 => 110000111110 => ? = 7
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ? => ? = 14
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ? => ? = 14
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => [7,7,1,1,1,1,1]
 => ? => ? = 7
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? => ? = 4

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000439

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 40%

Values

([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 7 = 6 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 6 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 8 = 7 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5 = 4 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5 = 4 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 6 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5 = 4 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 6 = 5 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4 = 3 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 8 = 7 + 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 + 1
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 6 + 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 8 = 7 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 7 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 14 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 7 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 6 = 5 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 6 = 5 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 5 = 4 + 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 5 = 4 + 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 5 = 4 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 5 = 4 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 5 = 4 + 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 8 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 5 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 6 = 5 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 7 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => ? = 6 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 8 = 7 + 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 7 = 6 + 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0]
 => ? = 10 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => 9 = 8 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 6 = 5 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => 8 = 7 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 5 = 4 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => 11 = 10 + 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => ? = 6 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 6 = 5 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => 11 = 10 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 9 = 8 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 7 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 8 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 5 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => 11 = 10 + 1
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 14 + 1
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 12 + 1
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 2 + 1
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ?
 => ? = 6 + 1
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 + 1
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 3 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ?
 => ? = 7 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 7 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 4 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ?
 => ? = 14 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 7 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 14 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 4 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [5,2,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 5 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [6,4,3,3]
 => [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ?
 => ? = 6 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,4,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
 => ? = 8 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 5 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 4 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 8 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 5 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => ? = 7 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [10,7]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ?
 => ? = 10 + 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ?
 => ? = 7 + 1
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ?
 => ? = 14 + 1

Description

The position of the first down step of a Dyck path.

Matching statistic: St000676
(load all 2 compositions to match this statistic)

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 28%

Values

([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 2
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 6
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 6
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 3
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 2
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? = 6
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 4
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 14
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 4
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 8
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 6
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 7
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 6
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 10
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 6
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 5
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,3),(1,2),(1,4),(3,4)],5)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 8
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 7
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 14
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 12
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 2
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ? = 6
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 6
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 6
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 4
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ? = 14
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 7
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 14
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [6,4,3,3]
 => [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,4,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 8
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 8
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 7
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [10,7]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 10
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 7

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: St001039
(load all 2 compositions to match this statistic)

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 28%

Values

([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 2
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 6
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 6
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 3
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 2
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? = 6
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 4
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 14
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 4
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 8
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 6
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 7
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 6
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 10
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 6
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 5
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,3),(1,2),(1,4),(3,4)],5)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 8
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 7
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 14
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 12
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 2
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ? = 6
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 6
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 6
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 4
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ? = 14
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 7
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 14
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [6,4,3,3]
 => [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,4,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 8
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 8
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 7
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [10,7]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 10
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 7

Description

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

Matching statistic: St000734

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%

Values

([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => 3
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 2
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([],4)
 => [2,2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
 => 2
([(2,3)],4)
 => [6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => 6
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 4
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 4
([(1,3),(2,3)],4)
 => [6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => 6
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => 3
([(0,3),(1,2)],4)
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 7
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]]
 => ? = 2
([(3,4)],5)
 => [6,6,6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
 => ? = 6
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
 => ? = 6
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
 => ? = 6
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
 => ? = 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12,13]]
 => ? = 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? = 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => 4
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
 => ? = 14
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? = 7
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [[1,2,3,4],[5,6,7],[8,9,10]]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => 4
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
 => 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => 4
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
 => ? = 6
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
 => ? = 6
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
 => ? = 3
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
 => ? = 6
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
 => ? = 8
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 5
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? = 7
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
 => ? = 6
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12,13]]
 => ? = 7
([(1,4),(2,3)],5)
 => [6,6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
 => ? = 6
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16]]
 => ? = 10
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11]]
 => ? = 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14]]
 => ? = 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => 4
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
 => ? = 6
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
 => ? = 6
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => 5
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [[1,2,3,4,5,6,7,8,9,10],[11,12]]
 => ? = 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? = 7
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
 => ? = 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
 => 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,3),(1,2),(1,4),(3,4)],5)
 => [8,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11]]
 => ? = 8
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => 7
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 5
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
 => ? = 14
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 2
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ? = 6
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 6
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 6
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 4
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ? = 14
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 7
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 14
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [6,4,3,3]
 => [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14,15,16]]
 => ? = 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,4,2]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12],[13,14]]
 => ? = 8
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
 => ? = 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [6,5,4]
 => [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15]]
 => ? = 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [8,2,2]
 => [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
 => ? = 8
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12,13],[14,15,16,17,18,19]]
 => ? = 7
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [10,7]
 => [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17]]
 => ? = 10
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11],[12,13],[14,15]]
 => ? = 7

Description

The last entry in the first row of a standard tableau.

The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000025The number of initial rises of a Dyck path. St000444The length of the maximal rise of a Dyck path. 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. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in 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. St000931The number of occurrences of the pattern UUU in a Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000738The first entry in the last row of a standard tableau. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one.