- FindStat

Your data matches 69 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000069
(load all 28 compositions to match this statistic)

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => 1
[2,1] => ([],2)
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => 1
[2,3,1] => ([(1,2)],3)
 => 2
[3,1,2] => ([(1,2)],3)
 => 2
[3,2,1] => ([],3)
 => 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 2
[2,4,3,1] => ([(1,2),(1,3)],4)
 => 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => 3
[4,1,2,3] => ([(1,2),(2,3)],4)
 => 2
[4,1,3,2] => ([(1,2),(1,3)],4)
 => 3
[4,2,1,3] => ([(1,3),(2,3)],4)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => 3
[4,3,1,2] => ([(2,3)],4)
 => 3
[4,3,2,1] => ([],4)
 => 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2

Description

The number of maximal elements of a poset.

Matching statistic: St000010
(load all 3 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [1]
 => 1
[1,2] => [2,1] => [2,1] => [2]
 => 1
[2,1] => [1,2] => [1,2] => [1,1]
 => 2
[1,2,3] => [3,2,1] => [3,1,2] => [3]
 => 1
[1,3,2] => [2,3,1] => [3,2,1] => [2,1]
 => 2
[2,1,3] => [3,1,2] => [2,3,1] => [3]
 => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1]
 => 2
[3,1,2] => [2,1,3] => [2,1,3] => [2,1]
 => 2
[3,2,1] => [1,2,3] => [1,2,3] => [1,1,1]
 => 3
[1,2,3,4] => [4,3,2,1] => [4,1,2,3] => [4]
 => 1
[1,2,4,3] => [3,4,2,1] => [4,1,3,2] => [3,1]
 => 2
[1,3,2,4] => [4,2,3,1] => [4,3,1,2] => [4]
 => 1
[1,3,4,2] => [2,4,3,1] => [4,2,1,3] => [3,1]
 => 2
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [2,2]
 => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [2,1,1]
 => 3
[2,1,3,4] => [4,3,1,2] => [2,4,1,3] => [4]
 => 1
[2,1,4,3] => [3,4,1,2] => [2,4,3,1] => [3,1]
 => 2
[2,3,1,4] => [4,1,3,2] => [3,4,2,1] => [4]
 => 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [3,1]
 => 2
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => [2,2]
 => 2
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
 => 3
[3,1,2,4] => [4,2,1,3] => [3,1,4,2] => [4]
 => 1
[3,1,4,2] => [2,4,1,3] => [3,2,4,1] => [3,1]
 => 2
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [4]
 => 1
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [3,1]
 => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,2]
 => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 3
[4,1,2,3] => [3,2,1,4] => [3,1,2,4] => [3,1]
 => 2
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [2,1,1]
 => 3
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [3,1]
 => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 3
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 3
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => [5]
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => [4,1]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,2,3] => [5]
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [5,1,3,2,4] => [4,1]
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => [5,1,4,3,2] => [3,2]
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [5,1,3,4,2] => [3,1,1]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,1,2,4] => [5]
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,1,4,2] => [4,1]
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,1,3,2] => [5]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [5,2,1,3,4] => [4,1]
 => 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,1,2,3] => [3,2]
 => 2
[1,3,5,4,2] => [2,4,5,3,1] => [5,2,1,4,3] => [3,1,1]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,2,1,3] => [5]
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,1,2] => [4,1]
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,3,4,1,2] => [5]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [5,2,4,1,3] => [4,1]
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [5,3,2,1,4] => [3,2]
 => 2
[8,7,3,4,5,6,1,2] => [2,1,6,5,4,3,7,8] => ? => ?
 => ? = 4
[8,4,5,6,7,2,1,3] => [3,1,2,7,6,5,4,8] => [2,3,1,7,4,5,6,8] => ?
 => ? = 3
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [3,1,2,6,4,5,7,8] => ?
 => ? = 4
[7,5,4,6,3,2,1,8] => [8,1,2,3,6,4,5,7] => [2,3,6,5,7,4,8,1] => ?
 => ? = 1
[7,6,4,3,5,2,1,8] => [8,1,2,5,3,4,6,7] => [2,5,4,6,3,7,8,1] => ?
 => ? = 1
[7,6,4,3,2,5,1,8] => [8,1,5,2,3,4,6,7] => [5,3,4,6,2,7,8,1] => ?
 => ? = 1
[2,1,6,5,8,7,4,3] => [3,4,7,8,5,6,1,2] => [2,8,3,4,6,1,7,5] => ?
 => ? = 4
[4,3,6,5,2,1,8,7] => [7,8,1,2,5,6,3,4] => [2,5,4,8,6,3,7,1] => ?
 => ? = 2
[1,6,5,7,4,3,2,8] => [8,2,3,4,7,5,6,1] => [8,3,4,7,6,1,5,2] => ?
 => ? = 1
[1,3,2,7,6,5,4,8] => [8,4,5,6,7,2,3,1] => ? => ?
 => ? = 1
[1,3,4,2,5,7,6,8] => [8,6,7,5,2,4,3,1] => [8,4,1,3,2,7,5,6] => ?
 => ? = 1
[1,8,7,4,6,5,3,2] => [2,3,5,6,4,7,8,1] => [8,2,3,6,5,4,7,1] => ?
 => ? = 6
[1,8,7,5,4,6,3,2] => [2,3,6,4,5,7,8,1] => [8,2,3,5,6,4,7,1] => ?
 => ? = 5
[1,8,7,5,6,4,3,2] => [2,3,4,6,5,7,8,1] => [8,2,3,4,6,5,7,1] => ?
 => ? = 6
[2,1,4,3,8,6,7,5] => [5,7,6,8,3,4,1,2] => [2,8,4,1,5,7,6,3] => ?
 => ? = 3
[2,1,8,5,6,7,4,3] => [3,4,7,6,5,8,1,2] => [2,8,3,4,7,5,6,1] => ?
 => ? = 4
[2,4,1,6,3,8,5,7] => [7,5,8,3,6,1,4,2] => [4,8,6,2,7,1,5,3] => ?
 => ? = 2
[1,2,5,3,6,4,7,8] => [8,7,4,6,3,5,2,1] => [8,1,5,6,2,3,4,7] => ?
 => ? = 1
[1,6,2,7,3,8,4,5] => [5,4,8,3,7,2,6,1] => [8,6,7,5,4,1,2,3] => ?
 => ? = 2
[1,2,3,6,4,7,5,8] => [8,5,7,4,6,3,2,1] => [8,1,2,6,7,3,4,5] => ?
 => ? = 1
[1,7,8,2,3,4,5,6] => [6,5,4,3,2,8,7,1] => [8,6,2,3,4,5,1,7] => ?
 => ? = 2
[2,4,1,6,3,8,5,9,7] => [7,9,5,8,3,6,1,4,2] => [4,9,6,2,8,1,7,3,5] => ?
 => ? = 2
[3,1,5,2,8,4,7,6] => [6,7,4,8,2,5,1,3] => [3,5,8,7,1,6,4,2] => ?
 => ? = 3
[2,1,7,8,5,6,4,3] => [3,4,6,5,8,7,1,2] => [2,8,3,4,6,5,1,7] => ?
 => ? = 4
[8,3,5,6,2,7,1,4] => [4,1,7,2,6,5,3,8] => [4,6,7,1,3,5,2,8] => ?
 => ? = 3
[6,5,2,1,4,3,8,7] => [7,8,3,4,1,2,5,6] => [2,5,4,1,6,8,7,3] => ?
 => ? = 2
[7,5,2,8,3,4,6,1] => [1,6,4,3,8,2,5,7] => [1,5,6,3,7,4,8,2] => ?
 => ? = 3
[1,4,5,6,7,8,9,2,3] => [3,2,9,8,7,6,5,4,1] => [9,3,2,1,4,5,6,7,8] => ?
 => ? = 2
[7,4,8,2,5,9,1,3,6,10] => [10,6,3,1,9,5,2,8,4,7] => [9,8,1,7,2,3,10,4,5,6] => ?
 => ? = 1
[1,8,7,5,4,3,6,2] => [2,6,3,4,5,7,8,1] => [8,2,4,5,6,3,7,1] => ?
 => ? = 4
[2,1,7,8,6,4,5,3] => [3,5,4,6,8,7,1,2] => [2,8,3,5,4,6,1,7] => ?
 => ? = 4
[8,7,6,4,5,3,2,1,9] => [9,1,2,3,5,4,6,7,8] => [2,3,5,6,4,7,8,9,1] => ?
 => ? = 1
[1,9,8,7,5,6,4,3,2] => [2,3,4,6,5,7,8,9,1] => [9,2,3,4,6,5,7,8,1] => ?
 => ? = 7
[9,8,7,5,6,4,3,2,1,10] => [10,1,2,3,4,6,5,7,8,9] => [2,3,4,6,7,5,8,9,10,1] => ?
 => ? = 1
[8,7,5,4,6,3,2,1,9] => [9,1,2,3,6,4,5,7,8] => [2,3,6,5,7,4,8,9,1] => ?
 => ? = 1
[8,7,4,6,5,3,2,1,9] => [9,1,2,3,5,6,4,7,8] => [2,3,5,7,6,4,8,9,1] => ?
 => ? = 1
[1,9,8,6,5,7,4,3,2] => [2,3,4,7,5,6,8,9,1] => [9,2,3,4,6,7,5,8,1] => ?
 => ? = 6
[1,9,8,5,7,6,4,3,2] => [2,3,4,6,7,5,8,9,1] => [9,2,3,4,7,6,5,8,1] => ?
 => ? = 7
[1,10,9,8,6,7,5,4,3,2] => [2,3,4,5,7,6,8,9,10,1] => [10,2,3,4,5,7,6,8,9,1] => ?
 => ? = 8
[6,3,7,1,4,8,2,5] => [5,2,8,4,1,7,3,6] => [7,5,6,1,2,8,3,4] => ?
 => ? = 2
[8,9,7,6,5,4,3,2,1,10] => [10,1,2,3,4,5,6,7,9,8] => [2,3,4,5,6,7,9,10,8,1] => ?
 => ? = 1
[1,8,7,9,6,5,4,3,2] => [2,3,4,5,6,9,7,8,1] => [9,2,3,4,5,6,8,1,7] => ?
 => ? = 6
[1,7,9,8,6,5,4,3,2] => [2,3,4,5,6,8,9,7,1] => [9,2,3,4,5,6,1,8,7] => ?
 => ? = 7
[8,7,6,3,2,5,4,1] => [1,4,5,2,3,6,7,8] => [1,3,5,4,2,6,7,8] => ?
 => ? = 6
[1,8,9,2,3,4,5,6,7] => [7,6,5,4,3,2,9,8,1] => [9,7,2,3,4,5,6,1,8] => ?
 => ? = 2
[1,8,2,9,3,4,5,6,7] => [7,6,5,4,3,9,2,8,1] => [9,8,7,3,4,5,6,1,2] => ?
 => ? = 2
[1,9,10,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,10,9,1] => [10,8,2,3,4,5,6,7,1,9] => ?
 => ? = 2
[6,2,4,1,5,3,8,7] => [7,8,3,5,1,4,2,6] => [4,6,5,2,1,8,7,3] => ?
 => ? = 2
[2,4,1,6,3,8,5,7,9] => [9,7,5,8,3,6,1,4,2] => [4,9,6,2,8,1,5,3,7] => ?
 => ? = 1
[2,4,1,6,3,8,5,10,7,9] => [9,7,10,5,8,3,6,1,4,2] => [4,10,6,2,8,1,9,3,7,5] => ?
 => ? = 2

Description

The length of the partition.

Matching statistic: St000340

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 94%●distinct values known / distinct values provided: 73%

Values

[1] => [1] => [1,0]
 => [1,0]
 => 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ? = 6 - 1
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 5 - 1
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 6 - 1
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ? = 6 - 1
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 4 - 1
[7,6,8,4,3,5,2,1] => [1,2,5,3,4,8,6,7] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? = 4 - 1
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? = 4 - 1
[7,8,3,4,5,6,2,1] => [1,2,6,5,4,3,8,7] => ?
 => ?
 => ? = 4 - 1
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
 => ? = 6 - 1
[5,6,7,8,4,2,3,1] => [1,3,2,4,8,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 4 - 1
[7,6,8,5,3,2,4,1] => [1,4,2,3,5,8,6,7] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 - 1
[5,6,7,8,3,2,4,1] => [1,4,2,3,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 3 - 1
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 - 1
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 3 - 1
[7,8,6,2,3,4,5,1] => [1,5,4,3,2,6,8,7] => ?
 => ?
 => ? = 4 - 1
[7,6,8,2,3,4,5,1] => [1,5,4,3,2,8,6,7] => ?
 => ?
 => ? = 3 - 1
[6,7,8,2,3,4,5,1] => [1,5,4,3,2,8,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 3 - 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 6 - 1
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
 => ? = 6 - 1
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 6 - 1
[5,6,7,8,4,3,1,2] => [2,1,3,4,8,7,6,5] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 4 - 1
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 6 - 1
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 6 - 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 3 - 1
[8,7,3,4,5,6,1,2] => [2,1,6,5,4,3,7,8] => ?
 => ?
 => ? = 4 - 1
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 - 1
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 4 - 1
[5,6,7,8,4,2,1,3] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => ? = 3 - 1
[8,4,5,6,7,2,1,3] => [3,1,2,7,6,5,4,8] => ?
 => ?
 => ? = 3 - 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 4 - 1
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? = 4 - 1
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => ? = 3 - 1
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 1
[8,4,5,6,7,1,2,3] => [3,2,1,7,6,5,4,8] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 - 1
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[5,6,7,8,3,2,1,4] => [4,1,2,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[6,7,5,8,2,3,1,4] => [4,1,3,2,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[7,5,6,8,2,3,1,4] => [4,1,3,2,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[5,6,7,8,2,3,1,4] => [4,1,3,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[7,5,6,8,3,1,2,4] => [4,2,1,3,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[5,6,7,8,3,1,2,4] => [4,2,1,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[6,5,7,8,2,1,3,4] => [4,3,1,2,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[5,6,7,8,2,1,3,4] => [4,3,1,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[7,8,6,5,1,2,3,4] => [4,3,2,1,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[8,6,7,5,1,2,3,4] => [4,3,2,1,5,7,6,8] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 4 - 1
[7,6,8,5,1,2,3,4] => [4,3,2,1,5,8,6,7] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 3 - 1
[6,7,8,5,1,2,3,4] => [4,3,2,1,5,8,7,6] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 3 - 1

Description

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns

$110$ and

$001$ .

Matching statistic: St000297

Mapping

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 73% ●values known / values provided: 91%●distinct values known / distinct values provided: 73%

Values

[1] => [1] =>  =>  => ? = 1 - 1
[1,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[2,1] => [2,1] => 1 => 1 => 1 = 2 - 1
[1,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,3,2] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[2,1,3] => [2,1,3] => 10 => 01 => 0 = 1 - 1
[2,3,1] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[3,1,2] => [3,1,2] => 01 => 10 => 1 = 2 - 1
[3,2,1] => [3,2,1] => 11 => 11 => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => 010 => 010 => 0 = 1 - 1
[1,3,4,2] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => 100 => 001 => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[2,3,1,4] => [1,3,2,4] => 010 => 010 => 0 = 1 - 1
[2,3,4,1] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => 001 => 100 => 1 = 2 - 1
[2,4,3,1] => [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[3,1,2,4] => [3,1,2,4] => 010 => 010 => 0 = 1 - 1
[3,1,4,2] => [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => 110 => 011 => 0 = 1 - 1
[3,2,4,1] => [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[3,4,1,2] => [2,4,1,3] => 001 => 100 => 1 = 2 - 1
[3,4,2,1] => [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[4,1,2,3] => [4,1,2,3] => 001 => 100 => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => 011 => 110 => 2 = 3 - 1
[4,2,1,3] => [4,2,1,3] => 101 => 101 => 1 = 2 - 1
[4,2,3,1] => [4,1,3,2] => 011 => 110 => 2 = 3 - 1
[4,3,1,2] => [4,3,1,2] => 011 => 110 => 2 = 3 - 1
[4,3,2,1] => [4,3,2,1] => 111 => 111 => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0100 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0010 => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,3,4,2,5] => [1,2,4,3,5] => 0010 => 0100 => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,5,4,2] => [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 0100 => 0 = 1 - 1
[1,4,2,5,3] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => 0110 => 0110 => 0 = 1 - 1
[1,4,3,5,2] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,4,5,2,3] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,4,5,3,2] => [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[7,8,5,6,4,3,2,1] => ? => ? => ? => ? = 6 - 1
[7,8,6,4,5,3,2,1] => ? => ? => ? => ? = 6 - 1
[8,6,7,4,5,3,2,1] => ? => ? => ? => ? = 6 - 1
[8,4,5,6,7,3,2,1] => ? => ? => ? => ? = 5 - 1
[7,8,6,5,3,4,2,1] => ? => ? => ? => ? = 6 - 1
[8,6,7,5,3,4,2,1] => ? => ? => ? => ? = 6 - 1
[7,6,8,4,3,5,2,1] => ? => ? => ? => ? = 4 - 1
[6,7,8,3,4,5,2,1] => ? => ? => ? => ? = 4 - 1
[8,7,3,4,5,6,2,1] => ? => ? => ? => ? = 5 - 1
[7,8,3,4,5,6,2,1] => ? => ? => ? => ? = 4 - 1
[7,8,6,5,4,2,3,1] => ? => ? => ? => ? = 6 - 1
[8,7,5,6,4,2,3,1] => ? => ? => ? => ? = 6 - 1
[8,7,6,4,5,2,3,1] => ? => ? => ? => ? = 6 - 1
[8,4,5,6,7,2,3,1] => ? => ? => ? => ? = 4 - 1
[7,6,8,5,3,2,4,1] => ? => ? => ? => ? = 4 - 1
[6,7,8,5,2,3,4,1] => ? => ? => ? => ? = 4 - 1
[8,7,6,2,3,4,5,1] => ? => ? => ? => ? = 5 - 1
[7,8,6,2,3,4,5,1] => ? => ? => ? => ? = 4 - 1
[8,6,7,2,3,4,5,1] => ? => ? => ? => ? = 4 - 1
[7,6,8,2,3,4,5,1] => ? => ? => ? => ? = 3 - 1
[6,7,8,2,3,4,5,1] => ? => ? => ? => ? = 3 - 1
[8,6,7,5,4,3,1,2] => ? => ? => ? => ? = 6 - 1
[8,7,5,6,4,3,1,2] => ? => ? => ? => ? = 6 - 1
[8,7,6,4,5,3,1,2] => ? => ? => ? => ? = 6 - 1
[8,7,6,5,3,4,1,2] => ? => ? => ? => ? = 6 - 1
[8,7,3,4,5,6,1,2] => ? => ? => ? => ? = 4 - 1
[8,4,5,6,7,2,1,3] => ? => ? => ? => ? = 3 - 1
[8,5,6,7,4,1,2,3] => ? => ? => ? => ? = 4 - 1
[8,7,4,5,6,1,2,3] => ? => ? => ? => ? = 4 - 1
[8,4,5,6,7,1,2,3] => ? => ? => ? => ? = 3 - 1
[6,7,5,8,2,3,1,4] => ? => ? => ? => ? = 2 - 1
[7,5,6,8,3,1,2,4] => ? => ? => ? => ? = 2 - 1
[6,7,5,4,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,5,6,4,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[6,5,7,4,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,6,4,5,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,5,4,6,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[6,5,4,7,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,6,5,3,4,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,6,4,3,5,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,5,4,3,6,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,6,5,4,2,3,1,8] => ? => ? => ? => ? = 1 - 1
[7,6,5,3,2,4,1,8] => ? => ? => ? => ? = 1 - 1
[7,6,4,3,2,5,1,8] => ? => ? => ? => ? = 1 - 1
[7,6,5,4,3,1,2,8] => ? => ? => ? => ? = 1 - 1
[7,6,5,4,2,1,3,8] => ? => ? => ? => ? = 1 - 1
[7,6,5,3,2,1,4,8] => ? => ? => ? => ? = 1 - 1
[3,4,2,1,5,6,7,8] => ? => ? => ? => ? = 1 - 1
[4,2,3,1,5,6,7,8] => ? => ? => ? => ? = 1 - 1

Description

The number of leading ones in a binary word.

Matching statistic: St000011
(load all 3 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 90%●distinct values known / distinct values provided: 82%

Values

[1] => [1] => [.,.]
 => [1,0]
 => 1
[1,2] => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1
[2,1] => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 2
[1,2,3] => [3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 1
[1,3,2] => [2,3,1] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2
[2,1,3] => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 1
[2,3,1] => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 2
[3,1,2] => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 3
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,2,4,3] => [3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,3,4,2] => [2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,4,2,3] => [3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,4,3,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 3
[2,1,3,4] => [4,3,1,2] => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,3,1,4] => [4,1,3,2] => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,3,4,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,4,3,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,1,2,4] => [4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1,4] => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 1
[3,2,4,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 2
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,4,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 3
[4,1,2,3] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 2
[4,1,3,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 3
[4,2,1,3] => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 3
[4,3,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 3
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,5,2,4] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,3,5,4,2] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 6
[8,4,5,6,7,3,2,1] => [1,2,3,7,6,5,4,8] => [.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 6
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 6
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[7,6,8,4,3,5,2,1] => [1,2,5,3,4,8,6,7] => [.,[.,[[.,[.,.]],[[.,[.,.]],.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 4
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [.,[.,[[[.,.],.],[[[.,.],.],.]]]]
 => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 4
[8,7,3,4,5,6,2,1] => [1,2,6,5,4,3,7,8] => [.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 5
[7,8,3,4,5,6,2,1] => [1,2,6,5,4,3,8,7] => ?
 => ?
 => ? = 4
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6
[8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6
[5,6,7,8,4,2,3,1] => [1,3,2,4,8,7,6,5] => [.,[[.,.],[.,[[[[.,.],.],.],.]]]]
 => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6
[8,4,5,6,7,2,3,1] => [1,3,2,7,6,5,4,8] => [.,[[.,.],[[[[.,.],.],.],[.,.]]]]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4
[8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 4
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
 => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
[8,7,6,2,3,4,5,1] => [1,5,4,3,2,6,7,8] => [.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 5
[7,8,6,2,3,4,5,1] => [1,5,4,3,2,6,8,7] => ?
 => ?
 => ? = 4
[8,6,7,2,3,4,5,1] => [1,5,4,3,2,7,6,8] => [.,[[[[.,.],.],.],[[.,.],[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 4
[7,6,8,2,3,4,5,1] => [1,5,4,3,2,8,6,7] => ?
 => ?
 => ? = 3
[6,7,8,2,3,4,5,1] => [1,5,4,3,2,8,7,6] => [.,[[[[.,.],.],.],[[[.,.],.],.]]]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[5,6,7,8,4,3,1,2] => [2,1,3,4,8,7,6,5] => [[.,.],[.,[.,[[[[.,.],.],.],.]]]]
 => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6
[8,7,3,4,5,6,1,2] => [2,1,6,5,4,3,7,8] => ?
 => ?
 => ? = 4
[8,4,5,6,7,2,1,3] => [3,1,2,7,6,5,4,8] => ?
 => ?
 => ? = 3
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 4
[6,7,5,8,2,3,1,4] => [4,1,3,2,8,5,7,6] => [[.,[[.,.],.]],[[.,[[.,.],.]],.]]
 => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ? = 2
[7,5,6,8,2,3,1,4] => [4,1,3,2,8,6,5,7] => [[.,[[.,.],.]],[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ? = 2
[5,6,7,8,2,3,1,4] => [4,1,3,2,8,7,6,5] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
 => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
 => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2
[7,5,6,8,3,1,2,4] => [4,2,1,3,8,6,5,7] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 2
[7,8,6,5,1,2,3,4] => [4,3,2,1,5,6,8,7] => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
[8,6,7,5,1,2,3,4] => [4,3,2,1,5,7,6,8] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4
[6,7,8,5,1,2,3,4] => [4,3,2,1,5,8,7,6] => [[[[.,.],.],.],[.,[[[.,.],.],.]]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
[8,6,5,7,1,2,3,4] => [4,3,2,1,7,5,6,8] => [[[[.,.],.],.],[[.,[.,.]],[.,.]]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
 => ? = 3
[8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[7,6,5,8,1,2,3,4] => [4,3,2,1,8,5,6,7] => [[[[.,.],.],.],[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[6,7,5,8,1,2,3,4] => [4,3,2,1,8,5,7,6] => [[[[.,.],.],.],[[.,[[.,.],.]],.]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
 => ? = 2
[7,5,6,8,1,2,3,4] => [4,3,2,1,8,6,5,7] => [[[[.,.],.],.],[[[.,.],[.,.]],.]]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
 => ? = 2
[7,5,4,6,3,2,1,8] => [8,1,2,3,6,4,5,7] => [[.,[.,[.,[[.,[.,.]],[.,.]]]]],.]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 1
[7,6,4,3,2,5,1,8] => [8,1,5,2,3,4,6,7] => [[.,[[.,[.,[.,.]]],[.,[.,.]]]],.]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1
[4,3,8,7,6,5,2,1] => [1,2,5,6,7,8,3,4] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6
[6,5,4,3,8,7,2,1] => [1,2,7,8,3,4,5,6] => [.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 4
[4,3,6,5,8,7,2,1] => [1,2,7,8,5,6,3,4] => [.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 4
[2,1,6,7,8,5,4,3] => [3,4,5,8,7,6,1,2] => [[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000745

Mapping

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 90%●distinct values known / distinct values provided: 73%

Values

[1] => [1] => [[1]]
 => [[1]]
 => 1
[1,2] => [1,2] => [[1,2]]
 => [[1,2]]
 => 1
[2,1] => [2,1] => [[1],[2]]
 => [[1],[2]]
 => 2
[1,2,3] => [1,2,3] => [[1,2,3]]
 => [[1,2,3]]
 => 1
[1,3,2] => [1,3,2] => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[3,1,2] => [3,1,2] => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
 => [[1],[2],[3]]
 => 3
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => [[1,2,3,4]]
 => 1
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 1
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => [[1,2,3],[4]]
 => 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 1
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 1
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 1
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
 => [[1,3],[2],[4]]
 => 2
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => 1
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 1
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 2
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 1
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 2
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 1
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => [[1,2,5],[3],[4]]
 => 1
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[7,8,5,6,4,3,2,1] => ? => ?
 => ?
 => ? = 6
[7,8,6,4,5,3,2,1] => ? => ?
 => ?
 => ? = 6
[8,6,7,4,5,3,2,1] => ? => ?
 => ?
 => ? = 6
[8,4,5,6,7,3,2,1] => ? => ?
 => ?
 => ? = 5
[7,8,6,5,3,4,2,1] => ? => ?
 => ?
 => ? = 6
[8,6,7,5,3,4,2,1] => ? => ?
 => ?
 => ? = 6
[7,6,8,4,3,5,2,1] => ? => ?
 => ?
 => ? = 4
[6,7,8,3,4,5,2,1] => ? => ?
 => ?
 => ? = 4
[8,7,3,4,5,6,2,1] => ? => ?
 => ?
 => ? = 5
[7,8,3,4,5,6,2,1] => ? => ?
 => ?
 => ? = 4
[7,8,6,5,4,2,3,1] => ? => ?
 => ?
 => ? = 6
[8,7,5,6,4,2,3,1] => ? => ?
 => ?
 => ? = 6
[5,6,7,8,4,2,3,1] => [1,2,4,8,7,3,6,5] => ?
 => ?
 => ? = 4
[8,7,6,4,5,2,3,1] => ? => ?
 => ?
 => ? = 6
[8,4,5,6,7,2,3,1] => ? => ?
 => ?
 => ? = 4
[7,6,8,5,3,2,4,1] => ? => ?
 => ?
 => ? = 4
[8,6,5,7,3,2,4,1] => [8,4,3,7,2,1,6,5] => ?
 => ?
 => ? = 4
[5,6,7,8,3,2,4,1] => [1,2,5,8,4,3,7,6] => ?
 => ?
 => ? = 3
[6,7,8,5,2,3,4,1] => ? => ?
 => ?
 => ? = 4
[8,5,6,7,2,3,4,1] => [8,2,4,7,1,3,6,5] => ?
 => ?
 => ? = 4
[8,7,6,2,3,4,5,1] => ? => ?
 => ?
 => ? = 5
[7,8,6,2,3,4,5,1] => ? => ?
 => ?
 => ? = 4
[8,6,7,2,3,4,5,1] => ? => ?
 => ?
 => ? = 4
[7,6,8,2,3,4,5,1] => ? => ?
 => ?
 => ? = 3
[6,7,8,2,3,4,5,1] => ? => ?
 => ?
 => ? = 3
[8,6,7,5,4,3,1,2] => ? => ?
 => ?
 => ? = 6
[8,7,5,6,4,3,1,2] => ? => ?
 => ?
 => ? = 6
[5,6,7,8,4,3,1,2] => [1,2,4,8,7,6,3,5] => ?
 => ?
 => ? = 4
[8,7,6,4,5,3,1,2] => ? => ?
 => ?
 => ? = 6
[8,7,6,5,3,4,1,2] => ? => ?
 => ?
 => ? = 6
[8,7,3,4,5,6,1,2] => ? => ?
 => ?
 => ? = 4
[7,8,3,4,5,6,1,2] => [5,8,1,2,4,7,3,6] => ?
 => ?
 => ? = 3
[8,4,5,6,7,2,1,3] => ? => ?
 => ?
 => ? = 3
[8,5,6,7,4,1,2,3] => ? => ?
 => ?
 => ? = 4
[8,7,4,5,6,1,2,3] => ? => ?
 => ?
 => ? = 4
[8,4,5,6,7,1,2,3] => ? => ?
 => ?
 => ? = 3
[6,7,5,8,2,3,1,4] => ? => ?
 => ?
 => ? = 2
[7,5,6,8,2,3,1,4] => [6,2,5,8,1,4,3,7] => ?
 => ?
 => ? = 2
[5,6,7,8,2,3,1,4] => [1,3,6,8,2,5,4,7] => ?
 => ?
 => ? = 2
[7,5,6,8,3,1,2,4] => ? => ?
 => ?
 => ? = 2
[5,6,7,8,3,1,2,4] => [1,3,6,8,5,2,4,7] => ?
 => ?
 => ? = 2
[6,5,7,8,2,1,3,4] => [4,3,6,8,2,1,5,7] => ?
 => ?
 => ? = 2
[7,6,8,5,1,2,3,4] => [5,4,8,7,1,2,3,6] => ?
 => ?
 => ? = 3
[7,8,5,6,1,2,3,4] => [5,8,4,7,1,2,3,6] => ?
 => ?
 => ? = 3
[8,6,5,7,1,2,3,4] => [8,5,4,7,1,2,3,6] => ?
 => ?
 => ? = 3
[7,6,5,8,1,2,3,4] => [6,5,4,8,1,2,3,7] => ?
 => ?
 => ? = 2
[6,7,5,8,1,2,3,4] => [3,6,5,8,1,2,4,7] => ?
 => ?
 => ? = 2
[6,5,7,8,1,2,3,4] => [4,3,6,8,1,2,5,7] => ?
 => ?
 => ? = 2
[6,7,5,4,3,2,1,8] => ? => ?
 => ?
 => ? = 1
[7,5,6,4,3,2,1,8] => ? => ?
 => ?
 => ? = 1

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

Matching statistic: St000925

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 89%●distinct values known / distinct values provided: 64%

Values

[1] => [1] => [1,0]
 => {{1}}
 => ? = 1
[1,2] => [2,1] => [1,1,0,0]
 => {{1,2}}
 => 1
[2,1] => [1,2] => [1,0,1,0]
 => {{1},{2}}
 => 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 2
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 3
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 2
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 3
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 3
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 3
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 3
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 2
[1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 3
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 8
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => {{1},{2},{3},{4},{5,6},{7,8}}
 => ? = 6
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3},{4},{5,6,7,8}}
 => ? = 5
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5},{6},{7,8}}
 => ? = 6
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => {{1},{2},{3},{4,5},{6,7},{8}}
 => ? = 6
[8,4,5,6,7,3,2,1] => [1,2,3,7,6,5,4,8] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2},{3},{4,5,6,7},{8}}
 => ? = 5
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4},{5},{6},{7,8}}
 => ? = 6
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5},{6,7},{8}}
 => ? = 6
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1},{2},{3,4},{5,6},{7},{8}}
 => ? = 6
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4},{5,6,7,8}}
 => ? = 4
[7,6,8,4,3,5,2,1] => [1,2,5,3,4,8,6,7] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5},{6,7,8}}
 => ? = 4
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5},{6,7,8}}
 => ? = 4
[8,7,3,4,5,6,2,1] => [1,2,6,5,4,3,7,8] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1},{2},{3,4,5,6},{7},{8}}
 => ? = 5
[7,8,3,4,5,6,2,1] => [1,2,6,5,4,3,8,7] => ?
 => ?
 => ? = 4
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => {{1},{2,3},{4},{5},{6},{7,8}}
 => ? = 6
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4},{5},{6,7},{8}}
 => ? = 6
[8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5,6},{7},{8}}
 => ? = 6
[5,6,7,8,4,2,3,1] => [1,3,2,4,8,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3},{4},{5,6,7,8}}
 => ? = 4
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => {{1},{2,3},{4,5},{6},{7},{8}}
 => ? = 6
[8,4,5,6,7,2,3,1] => [1,3,2,7,6,5,4,8] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2,3},{4,5,6,7},{8}}
 => ? = 4
[7,6,8,5,3,2,4,1] => [1,4,2,3,5,8,6,7] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => {{1},{2,3,4},{5},{6,7,8}}
 => ? = 4
[8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5,6,7},{8}}
 => ? = 4
[5,6,7,8,3,2,4,1] => [1,4,2,3,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4},{5,6,7,8}}
 => ? = 3
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => {{1},{2,3,4},{5},{6,7,8}}
 => ? = 4
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5,6,7},{8}}
 => ? = 4
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4},{5,6,7,8}}
 => ? = 3
[8,7,6,2,3,4,5,1] => [1,5,4,3,2,6,7,8] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => {{1},{2,3,4,5},{6},{7},{8}}
 => ? = 5
[7,8,6,2,3,4,5,1] => [1,5,4,3,2,6,8,7] => ?
 => ?
 => ? = 4
[8,6,7,2,3,4,5,1] => [1,5,4,3,2,7,6,8] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => {{1},{2,3,4,5},{6,7},{8}}
 => ? = 4
[7,6,8,2,3,4,5,1] => [1,5,4,3,2,8,6,7] => ?
 => ?
 => ? = 3
[6,7,8,2,3,4,5,1] => [1,5,4,3,2,8,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => {{1},{2,3,4,5},{6,7,8}}
 => ? = 3
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5},{6},{7,8}}
 => ? = 6
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4},{5},{6,7},{8}}
 => ? = 6
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4},{5,6},{7},{8}}
 => ? = 6
[5,6,7,8,4,3,1,2] => [2,1,3,4,8,7,6,5] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3},{4},{5,6,7,8}}
 => ? = 4
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4,5},{6},{7},{8}}
 => ? = 6
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 6
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 4
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4},{5,6,7,8}}
 => ? = 3
[8,7,3,4,5,6,1,2] => [2,1,6,5,4,3,7,8] => ?
 => ?
 => ? = 4
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2},{3,4,5,6},{7,8}}
 => ? = 3
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 4
[5,6,7,8,4,2,1,3] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 3
[8,4,5,6,7,2,1,3] => [3,1,2,7,6,5,4,8] => ?
 => ?
 => ? = 3
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 4
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => {{1,2,3},{4},{5,6,7},{8}}
 => ? = 4
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 3
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 4
[8,4,5,6,7,1,2,3] => [3,2,1,7,6,5,4,8] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3},{4,5,6,7},{8}}
 => ? = 3

Description

The number of topologically connected components of a set partition. For example, the set partition

$\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components

$\{1,4,5,6\}$ and

$\{2,3\}$ . The number of set partitions with only one block is [[oeis:A099947]].

Matching statistic: St001068
(load all 12 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 85%●distinct values known / distinct values provided: 64%

Values

[1] => [1] => [1,0]
 => 1
[1,2] => [2,1] => [1,1,0,0]
 => 1
[2,1] => [1,2] => [1,0,1,0]
 => 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 3
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 6
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 6
[8,4,5,6,7,3,2,1] => [1,2,3,7,6,5,4,8] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 6
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 6
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 6
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[7,6,8,4,3,5,2,1] => [1,2,5,3,4,8,6,7] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 4
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 4
[8,7,3,4,5,6,2,1] => [1,2,6,5,4,3,7,8] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 5
[7,8,3,4,5,6,2,1] => [1,2,6,5,4,3,8,7] => ?
 => ? = 4
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6
[8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6
[5,6,7,8,4,2,3,1] => [1,3,2,4,8,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6
[8,4,5,6,7,2,3,1] => [1,3,2,7,6,5,4,8] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4
[7,6,8,5,3,2,4,1] => [1,4,2,3,5,8,6,7] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
[5,6,7,8,3,2,4,1] => [1,4,2,3,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[8,7,6,2,3,4,5,1] => [1,5,4,3,2,6,7,8] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 5
[7,8,6,2,3,4,5,1] => [1,5,4,3,2,6,8,7] => ?
 => ? = 4
[8,6,7,2,3,4,5,1] => [1,5,4,3,2,7,6,8] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 4
[7,6,8,2,3,4,5,1] => [1,5,4,3,2,8,6,7] => ?
 => ? = 3
[6,7,8,2,3,4,5,1] => [1,5,4,3,2,8,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6
[5,6,7,8,4,3,1,2] => [2,1,3,4,8,7,6,5] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[8,7,3,4,5,6,1,2] => [2,1,6,5,4,3,7,8] => ?
 => ? = 4
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[5,6,7,8,4,2,1,3] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[8,4,5,6,7,2,1,3] => [3,1,2,7,6,5,4,8] => ?
 => ? = 3
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 4
[8,4,5,6,7,1,2,3] => [3,2,1,7,6,5,4,8] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St000053
(load all 12 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 85%●distinct values known / distinct values provided: 64%

Values

[1] => [1] => [1,0]
 => 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0]
 => 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
 => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1 = 2 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 - 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 6 - 1
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5 - 1
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 - 1
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 6 - 1
[8,4,5,6,7,3,2,1] => [1,2,3,7,6,5,4,8] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5 - 1
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 6 - 1
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 6 - 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 6 - 1
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[7,6,8,4,3,5,2,1] => [1,2,5,3,4,8,6,7] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 4 - 1
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 4 - 1
[8,7,3,4,5,6,2,1] => [1,2,6,5,4,3,7,8] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 5 - 1
[7,8,3,4,5,6,2,1] => [1,2,6,5,4,3,8,7] => ?
 => ? = 4 - 1
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 - 1
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6 - 1
[8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6 - 1
[5,6,7,8,4,2,3,1] => [1,3,2,4,8,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[8,4,5,6,7,2,3,1] => [1,3,2,7,6,5,4,8] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4 - 1
[7,6,8,5,3,2,4,1] => [1,4,2,3,5,8,6,7] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4 - 1
[5,6,7,8,3,2,4,1] => [1,4,2,3,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4 - 1
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[8,7,6,2,3,4,5,1] => [1,5,4,3,2,6,7,8] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 5 - 1
[7,8,6,2,3,4,5,1] => [1,5,4,3,2,6,8,7] => ?
 => ? = 4 - 1
[8,6,7,2,3,4,5,1] => [1,5,4,3,2,7,6,8] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[7,6,8,2,3,4,5,1] => [1,5,4,3,2,8,6,7] => ?
 => ? = 3 - 1
[6,7,8,2,3,4,5,1] => [1,5,4,3,2,8,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 - 1
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6 - 1
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6 - 1
[5,6,7,8,4,3,1,2] => [2,1,3,4,8,7,6,5] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[8,7,3,4,5,6,1,2] => [2,1,6,5,4,3,7,8] => ?
 => ? = 4 - 1
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 - 1
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[5,6,7,8,4,2,1,3] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[8,4,5,6,7,2,1,3] => [3,1,2,7,6,5,4,8] => ?
 => ? = 3 - 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4 - 1
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 4 - 1
[8,4,5,6,7,1,2,3] => [3,2,1,7,6,5,4,8] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3 - 1
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1

Description

The number of valleys of the Dyck path.

Matching statistic: St000105
(load all 4 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 85%●distinct values known / distinct values provided: 64%

Values

[1] => [1] => [1] => {{1}}
 => 1
[1,2] => [2,1] => [2,1] => {{1,2}}
 => 1
[2,1] => [1,2] => [1,2] => {{1},{2}}
 => 2
[1,2,3] => [3,2,1] => [2,3,1] => {{1,2,3}}
 => 1
[1,3,2] => [2,3,1] => [3,2,1] => {{1,3},{2}}
 => 2
[2,1,3] => [3,1,2] => [3,1,2] => {{1,2,3}}
 => 1
[2,3,1] => [1,3,2] => [1,3,2] => {{1},{2,3}}
 => 2
[3,1,2] => [2,1,3] => [2,1,3] => {{1,2},{3}}
 => 2
[3,2,1] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
 => 3
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => {{1,2,3,4}}
 => 1
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => {{1,2,4},{3}}
 => 2
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => {{1,2,3,4}}
 => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => {{1,3,4},{2}}
 => 2
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
 => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
 => 3
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => {{1,2,3,4}}
 => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => {{1,2,4},{3}}
 => 2
[2,3,1,4] => [4,1,3,2] => [4,3,1,2] => {{1,2,3,4}}
 => 1
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
 => 2
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => {{1,3},{2,4}}
 => 2
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
 => 3
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => {{1,2,3,4}}
 => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
 => 2
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => {{1,2,3,4}}
 => 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => {{1},{2,3,4}}
 => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
 => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 3
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
 => 2
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
 => 3
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => {{1,2,3},{4}}
 => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 3
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
 => 3
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,3,5,4,1] => {{1,2,3,5},{4}}
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [2,4,5,3,1] => {{1,2,3,4,5}}
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,4,3,5,1] => {{1,2,4,5},{3}}
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => {{1,2,5},{3,4}}
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [3,4,2,5,1] => {{1,2,3,4,5}}
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [3,5,2,4,1] => {{1,2,3,5},{4}}
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [3,4,5,1,2] => {{1,2,3,4,5}}
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [3,2,4,5,1] => {{1,3,4,5},{2}}
 => 2
[1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,1,2] => {{1,3,4},{2,5}}
 => 2
[1,3,5,4,2] => [2,4,5,3,1] => [3,2,5,4,1] => {{1,3,5},{2},{4}}
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => {{1,2,3,4,5}}
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => {{1,2,4,5},{3}}
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [4,5,2,3,1] => {{1,2,3,4,5}}
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [4,2,5,3,1] => {{1,3,4,5},{2}}
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,2,5,1] => {{1,4,5},{2,3}}
 => 2
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 8
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,2,3,4,6,5,8,7] => {{1},{2},{3},{4},{5,6},{7,8}}
 => ? = 6
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,2,3,4,6,7,8,5] => {{1},{2},{3},{4},{5,6,7,8}}
 => ? = 5
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,2,3,5,4,6,8,7] => {{1},{2},{3},{4,5},{6},{7,8}}
 => ? = 6
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
 => ? = 6
[8,4,5,6,7,3,2,1] => [1,2,3,7,6,5,4,8] => [1,2,3,5,6,7,4,8] => {{1},{2},{3},{4,5,6,7},{8}}
 => ? = 5
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,2,4,3,5,6,8,7] => {{1},{2},{3,4},{5},{6},{7,8}}
 => ? = 6
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [1,2,4,3,5,7,6,8] => {{1},{2},{3,4},{5},{6,7},{8}}
 => ? = 6
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
 => ? = 6
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,2,4,3,6,7,8,5] => {{1},{2},{3,4},{5,6,7,8}}
 => ? = 4
[7,6,8,4,3,5,2,1] => [1,2,5,3,4,8,6,7] => [1,2,5,3,4,8,6,7] => {{1},{2},{3,4,5},{6,7,8}}
 => ? = 4
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [1,2,4,5,3,7,8,6] => {{1},{2},{3,4,5},{6,7,8}}
 => ? = 4
[8,7,3,4,5,6,2,1] => [1,2,6,5,4,3,7,8] => [1,2,4,5,6,3,7,8] => {{1},{2},{3,4,5,6},{7},{8}}
 => ? = 5
[7,8,3,4,5,6,2,1] => [1,2,6,5,4,3,8,7] => [1,2,4,5,6,3,8,7] => {{1},{2},{3,4,5,6},{7,8}}
 => ? = 4
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => [1,3,2,4,5,6,8,7] => {{1},{2,3},{4},{5},{6},{7,8}}
 => ? = 6
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
 => ? = 6
[8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => [1,3,2,4,6,5,7,8] => {{1},{2,3},{4},{5,6},{7},{8}}
 => ? = 6
[5,6,7,8,4,2,3,1] => [1,3,2,4,8,7,6,5] => [1,3,2,4,6,7,8,5] => {{1},{2,3},{4},{5,6,7,8}}
 => ? = 4
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
 => ? = 6
[8,4,5,6,7,2,3,1] => [1,3,2,7,6,5,4,8] => [1,3,2,5,6,7,4,8] => {{1},{2,3},{4,5,6,7},{8}}
 => ? = 4
[7,6,8,5,3,2,4,1] => [1,4,2,3,5,8,6,7] => [1,4,2,3,5,8,6,7] => {{1},{2,3,4},{5},{6,7,8}}
 => ? = 4
[8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [1,4,2,3,7,5,6,8] => {{1},{2,3,4},{5,6,7},{8}}
 => ? = 4
[5,6,7,8,3,2,4,1] => [1,4,2,3,8,7,6,5] => [1,4,2,3,6,7,8,5] => {{1},{2,3,4},{5,6,7,8}}
 => ? = 3
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [1,3,4,2,5,7,8,6] => {{1},{2,3,4},{5},{6,7,8}}
 => ? = 4
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,3,4,2,6,7,5,8] => {{1},{2,3,4},{5,6,7},{8}}
 => ? = 4
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,3,4,2,6,7,8,5] => {{1},{2,3,4},{5,6,7,8}}
 => ? = 3
[8,7,6,2,3,4,5,1] => [1,5,4,3,2,6,7,8] => [1,3,4,5,2,6,7,8] => {{1},{2,3,4,5},{6},{7},{8}}
 => ? = 5
[7,8,6,2,3,4,5,1] => [1,5,4,3,2,6,8,7] => [1,3,4,5,2,6,8,7] => ?
 => ? = 4
[8,6,7,2,3,4,5,1] => [1,5,4,3,2,7,6,8] => [1,3,4,5,2,7,6,8] => {{1},{2,3,4,5},{6,7},{8}}
 => ? = 4
[7,6,8,2,3,4,5,1] => [1,5,4,3,2,8,6,7] => [1,3,4,5,2,8,6,7] => {{1},{2,3,4,5},{6,7,8}}
 => ? = 3
[6,7,8,2,3,4,5,1] => [1,5,4,3,2,8,7,6] => [1,3,4,5,2,7,8,6] => {{1},{2,3,4,5},{6,7,8}}
 => ? = 3
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,8,7] => {{1,2},{3},{4},{5},{6},{7,8}}
 => ? = 6
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => [2,1,3,4,5,7,6,8] => {{1,2},{3},{4},{5},{6,7},{8}}
 => ? = 6
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => [2,1,3,4,6,5,7,8] => {{1,2},{3},{4},{5,6},{7},{8}}
 => ? = 6
[5,6,7,8,4,3,1,2] => [2,1,3,4,8,7,6,5] => [2,1,3,4,6,7,8,5] => {{1,2},{3},{4},{5,6,7,8}}
 => ? = 4
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [2,1,3,5,4,6,7,8] => {{1,2},{3},{4,5},{6},{7},{8}}
 => ? = 6
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => [2,1,4,3,5,6,7,8] => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 6
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 4
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [2,1,4,3,6,7,8,5] => {{1,2},{3,4},{5,6,7,8}}
 => ? = 3
[8,7,3,4,5,6,1,2] => [2,1,6,5,4,3,7,8] => ? => ?
 => ? = 4
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [2,1,4,5,6,3,8,7] => {{1,2},{3,4,5,6},{7,8}}
 => ? = 3
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [3,1,2,4,5,8,6,7] => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 4
[5,6,7,8,4,2,1,3] => [3,1,2,4,8,7,6,5] => [3,1,2,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
 => ? = 3
[8,4,5,6,7,2,1,3] => [3,1,2,7,6,5,4,8] => [3,1,2,5,6,7,4,8] => {{1,2,3},{4,5,6,7},{8}}
 => ? = 3
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [2,3,1,4,5,7,8,6] => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 4
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
 => ? = 4
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
 => ? = 3
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [2,3,1,5,6,4,7,8] => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 4
[8,4,5,6,7,1,2,3] => [3,2,1,7,6,5,4,8] => [2,3,1,5,6,7,4,8] => {{1,2,3},{4,5,6,7},{8}}
 => ? = 3
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [4,1,2,3,8,5,6,7] => {{1,2,3,4},{5,6,7,8}}
 => ? = 2

Description

The number of blocks in the set partition. The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]]

$S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of

$\{ 1,\ldots,n\}$ into

$k$ blocks, see [1].

The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000678The number of up steps after the last double rise of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000167The number of leaves of an ordered tree. St000912The number of maximal antichains in a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000068The number of minimal elements in a poset. St000740The last entry of a permutation. St000654The first descent of a permutation. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000989The number of final rises of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000054The first entry of the permutation. St000990The first ascent of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St000015The number of peaks of a Dyck path. St000133The "bounce" of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000239The number of small weak excedances. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000061The number of nodes on the left branch of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. 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. St000702The number of weak deficiencies of a permutation. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.