Your data matches 36 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000256
(load all 5 compositions to match this statistic)
Mapping
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 1
[2,1]
 => 0
[1,1,1]
 => 0
[4]
 => 1
[3,1]
 => 1
[2,2]
 => 1
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 1
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 1
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
[6]
 => 1
[5,1]
 => 1
[4,2]
 => 2
[4,1,1]
 => 1
[3,3]
 => 1
[3,2,1]
 => 0
[3,1,1,1]
 => 1
[2,2,2]
 => 1
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 0
[7]
 => 1
[6,1]
 => 1
[5,2]
 => 2
[5,1,1]
 => 1
[4,3]
 => 1
[4,2,1]
 => 1
[4,1,1,1]
 => 1
[3,3,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 0
[8]
 => 1
[7,1]
 => 1
[6,2]
 => 2
[6,1,1]
 => 1
[5,3]
 => 2
[5,2,1]
 => 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000257
(load all 2 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0
[2]
 => [1,1]
 => 1
[1,1]
 => [2]
 => 0
[3]
 => [1,1,1]
 => 1
[2,1]
 => [2,1]
 => 0
[1,1,1]
 => [3]
 => 0
[4]
 => [1,1,1,1]
 => 1
[3,1]
 => [2,1,1]
 => 1
[2,2]
 => [2,2]
 => 1
[2,1,1]
 => [3,1]
 => 0
[1,1,1,1]
 => [4]
 => 0
[5]
 => [1,1,1,1,1]
 => 1
[4,1]
 => [2,1,1,1]
 => 1
[3,2]
 => [2,2,1]
 => 1
[3,1,1]
 => [3,1,1]
 => 1
[2,2,1]
 => [3,2]
 => 0
[2,1,1,1]
 => [4,1]
 => 0
[1,1,1,1,1]
 => [5]
 => 0
[6]
 => [1,1,1,1,1,1]
 => 1
[5,1]
 => [2,1,1,1,1]
 => 1
[4,2]
 => [2,2,1,1]
 => 2
[4,1,1]
 => [3,1,1,1]
 => 1
[3,3]
 => [2,2,2]
 => 1
[3,2,1]
 => [3,2,1]
 => 0
[3,1,1,1]
 => [4,1,1]
 => 1
[2,2,2]
 => [3,3]
 => 1
[2,2,1,1]
 => [4,2]
 => 0
[2,1,1,1,1]
 => [5,1]
 => 0
[1,1,1,1,1,1]
 => [6]
 => 0
[7]
 => [1,1,1,1,1,1,1]
 => 1
[6,1]
 => [2,1,1,1,1,1]
 => 1
[5,2]
 => [2,2,1,1,1]
 => 2
[5,1,1]
 => [3,1,1,1,1]
 => 1
[4,3]
 => [2,2,2,1]
 => 1
[4,2,1]
 => [3,2,1,1]
 => 1
[4,1,1,1]
 => [4,1,1,1]
 => 1
[3,3,1]
 => [3,2,2]
 => 1
[3,2,2]
 => [3,3,1]
 => 1
[3,2,1,1]
 => [4,2,1]
 => 0
[3,1,1,1,1]
 => [5,1,1]
 => 1
[2,2,2,1]
 => [4,3]
 => 0
[2,2,1,1,1]
 => [5,2]
 => 0
[2,1,1,1,1,1]
 => [6,1]
 => 0
[1,1,1,1,1,1,1]
 => [7]
 => 0
[8]
 => [1,1,1,1,1,1,1,1]
 => 1
[7,1]
 => [2,1,1,1,1,1,1]
 => 1
[6,2]
 => [2,2,1,1,1,1]
 => 2
[6,1,1]
 => [3,1,1,1,1,1]
 => 1
[5,3]
 => [2,2,2,1,1]
 => 2
[5,2,1]
 => [3,2,1,1,1]
 => 1
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St000386
(load all 32 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001092
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1]
 => 0
[2]
 => [1,1]
 => [2]
 => 1
[1,1]
 => [2]
 => [1,1]
 => 0
[3]
 => [1,1,1]
 => [2,1]
 => 1
[2,1]
 => [2,1]
 => [1,1,1]
 => 0
[1,1,1]
 => [3]
 => [3]
 => 0
[4]
 => [1,1,1,1]
 => [2,2]
 => 1
[3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[2,2]
 => [2,2]
 => [4]
 => 1
[2,1,1]
 => [3,1]
 => [3,1]
 => 0
[1,1,1,1]
 => [4]
 => [1,1,1,1]
 => 0
[5]
 => [1,1,1,1,1]
 => [2,2,1]
 => 1
[4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[3,2]
 => [2,2,1]
 => [4,1]
 => 1
[3,1,1]
 => [3,1,1]
 => [3,2]
 => 1
[2,2,1]
 => [3,2]
 => [3,1,1]
 => 0
[2,1,1,1]
 => [4,1]
 => [1,1,1,1,1]
 => 0
[1,1,1,1,1]
 => [5]
 => [5]
 => 0
[6]
 => [1,1,1,1,1,1]
 => [2,2,2]
 => 1
[5,1]
 => [2,1,1,1,1]
 => [2,2,1,1]
 => 1
[4,2]
 => [2,2,1,1]
 => [4,2]
 => 2
[4,1,1]
 => [3,1,1,1]
 => [3,2,1]
 => 1
[3,3]
 => [2,2,2]
 => [4,1,1]
 => 1
[3,2,1]
 => [3,2,1]
 => [3,1,1,1]
 => 0
[3,1,1,1]
 => [4,1,1]
 => [2,1,1,1,1]
 => 1
[2,2,2]
 => [3,3]
 => [6]
 => 1
[2,2,1,1]
 => [4,2]
 => [1,1,1,1,1,1]
 => 0
[2,1,1,1,1]
 => [5,1]
 => [5,1]
 => 0
[1,1,1,1,1,1]
 => [6]
 => [3,3]
 => 0
[7]
 => [1,1,1,1,1,1,1]
 => [2,2,2,1]
 => 1
[6,1]
 => [2,1,1,1,1,1]
 => [2,2,1,1,1]
 => 1
[5,2]
 => [2,2,1,1,1]
 => [4,2,1]
 => 2
[5,1,1]
 => [3,1,1,1,1]
 => [3,2,2]
 => 1
[4,3]
 => [2,2,2,1]
 => [4,1,1,1]
 => 1
[4,2,1]
 => [3,2,1,1]
 => [3,2,1,1]
 => 1
[4,1,1,1]
 => [4,1,1,1]
 => [2,1,1,1,1,1]
 => 1
[3,3,1]
 => [3,2,2]
 => [4,3]
 => 1
[3,2,2]
 => [3,3,1]
 => [6,1]
 => 1
[3,2,1,1]
 => [4,2,1]
 => [1,1,1,1,1,1,1]
 => 0
[3,1,1,1,1]
 => [5,1,1]
 => [5,2]
 => 1
[2,2,2,1]
 => [4,3]
 => [3,1,1,1,1]
 => 0
[2,2,1,1,1]
 => [5,2]
 => [5,1,1]
 => 0
[2,1,1,1,1,1]
 => [6,1]
 => [3,3,1]
 => 0
[1,1,1,1,1,1,1]
 => [7]
 => [7]
 => 0
[8]
 => [1,1,1,1,1,1,1,1]
 => [2,2,2,2]
 => 1
[7,1]
 => [2,1,1,1,1,1,1]
 => [2,2,2,1,1]
 => 1
[6,2]
 => [2,2,1,1,1,1]
 => [4,2,2]
 => 2
[6,1,1]
 => [3,1,1,1,1,1]
 => [3,2,2,1]
 => 1
[5,3]
 => [2,2,2,1,1]
 => [4,2,1,1]
 => 2
[5,2,1]
 => [3,2,1,1,1]
 => [3,2,1,1,1]
 => 1
[]
 => []
 => ?
 => ? = 0
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St000201
(load all 7 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [.,[.,.]]
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,.]]
 => 2 = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => 2 = 1 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => 1 = 0 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => 2 = 1 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[.,.]],[.,.]]
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => 2 = 1 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 2 = 1 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 1 = 0 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => 1 = 0 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[.,.],.],.],.],.],.],[.,.]]
 => 2 = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[[[[.,[.,.]],.],.],.],[.,.]]
 => 2 = 1 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[[.,[[.,.],.]],.],[.,.]]
 => 2 = 1 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2 = 1 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 1 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => 2 = 1 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2 = 1 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 1 = 0 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[[.,.],[.,.]],.],.]]
 => 2 = 1 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 1 = 0 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => 1 = 0 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[[[.,[.,.]],.],.],.],.]]
 => 1 = 0 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[[[[[.,.],.],.],.],.],.],.]]
 => 1 = 0 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
 => 2 = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
 => 2 = 1 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[.,[[.,.],.]],.],.],[.,.]]
 => 2 = 1 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => 2 = 1 + 1
[]
 => []
 => .
 => ? = 0 + 1
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000291
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => 0 => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 01 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 110 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 00 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 011 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1110 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0111 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 11110 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 0110 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 0011 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 01111 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => 111110 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,3,2,1,6] => 01110 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1010 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 0110 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1101 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 0101 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1011 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 0011 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,4,3,2] => 00111 => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => 011111 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => 1111110 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,6,4,3,2,1,7] => 011110 => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,3,5,2,1,6] => 10110 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,5,4,2,1,6] => 01110 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1100 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0010 => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0110 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 0
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,4,6,3,2] => 01011 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0011 => 0
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,5,3,2] => 00111 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,5,4,3,2] => 001111 => 0
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,7,6,5,4,3,2] => 0111111 => 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,7,6,5,4,3,2,1,9] => 11111110 => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [6,7,5,4,3,2,1,8] => 0111110 => 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [5,4,6,3,2,1,7] => 101110 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [4,6,5,3,2,1,7] => 011110 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => 11010 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,2,1,6] => 00110 => 1
[]
 => []
 => [] => ? => ? = 0
Description
The number of descents of a binary word.
Matching statistic: St001712
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => [[1,2]]
 => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => [[1,2,3]]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [[1,6],[2],[3],[4],[5]]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [[1,2,5],[3],[4]]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [[1,2,3],[4],[5]]
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => [[1,7],[2],[3],[4],[5],[6]]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,3,2,1,6] => [[1,2,6],[3],[4],[5]]
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [[1,3,5],[2],[4]]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [[1,2,5],[3],[4]]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [[1,4],[2,5],[3]]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [[1,2,4],[3],[5]]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [[1,2,3],[4],[5]]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,4,3,2] => [[1,2,3],[4],[5],[6]]
 => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7]]
 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => [[1,8],[2],[3],[4],[5],[6],[7]]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,6,4,3,2,1,7] => [[1,2,7],[3],[4],[5],[6]]
 => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,3,5,2,1,6] => [[1,3,6],[2],[4],[5]]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,5,4,2,1,6] => [[1,2,6],[3],[4],[5]]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [[1,4,5],[2],[3]]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => 0
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,4,6,3,2] => [[1,2,4],[3],[5],[6]]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 0
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,5,3,2] => [[1,2,3],[4],[5],[6]]
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,5,4,3,2] => [[1,2,3],[4],[5],[6],[7]]
 => 0
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8]]
 => 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,7,6,5,4,3,2,1,9] => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
 => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [6,7,5,4,3,2,1,8] => [[1,2,8],[3],[4],[5],[6],[7]]
 => 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [5,4,6,3,2,1,7] => [[1,3,7],[2],[4],[5],[6]]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [4,6,5,3,2,1,7] => [[1,2,7],[3],[4],[5],[6]]
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => [[1,4,6],[2],[3],[5]]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,2,1,6] => [[1,2,3,6],[4],[5]]
 => 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,4,3,1,6] => [[1,2,6],[3],[4],[5]]
 => 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
 => ? = 0
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,8,7,6,5,4,3,2,1,10] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [7,8,6,5,4,3,2,1,9] => [[1,2,9],[3],[4],[5],[6],[7],[8]]
 => ? = 1
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,8,9,7,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
 => ? = 0
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 0
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [8,9,7,6,5,4,3,2,1,10] => [[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
 => ? = 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
 => ? = 2
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [6,8,7,5,4,3,2,1,9] => [[1,2,9],[3],[4],[5],[6],[7],[8]]
 => ? = 1
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,8,7,9,6,5,4,3,2] => [[1,2,4],[3],[5],[6],[7],[8],[9]]
 => ? = 1
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,7,9,8,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
 => ? = 0
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,9,10,8,7,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 0
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 0
[]
 => []
 => [] => []
 => ? = 0
Description
The number of natural descents of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St000071
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[.,.],.],.],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[[[[.,[.,.]],.],.],.],[.,.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => 2 = 1 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[[.,[[.,.],.]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[[.,.],[.,.]],.],.]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 2 = 1 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[[[.,[.,.]],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[[[[[.,.],.],.],.],.],.],.]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1 = 0 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[.,[[.,.],.]],.],.],[.,.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => 2 = 1 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 2 = 1 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
 => ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
 => ? = 1 + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 1 + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
 => ? = 2 + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[[[[.,[[.,.],.]],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
 => ([(0,10),(1,9),(2,10),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],11)
 => ? = 1 + 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
 => ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
 => ? = 1 + 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,4),(6,3),(8,5)],9)
 => ? = 2 + 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 1 + 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
 => ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
 => ? = 2 + 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [[[[.,[[[.,.],.],.]],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 0 + 1
[]
 => []
 => .
 => ?
 => ? = 0 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000068
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[.,.],.],.],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[[[[.,[.,.]],.],.],.],[.,.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => 2 = 1 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[[.,[[.,.],.]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[[.,.],[.,.]],.],.]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 2 = 1 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[[[.,[.,.]],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[[[[[.,.],.],.],.],.],.],.]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1 = 0 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[.,[[.,.],.]],.],.],[.,.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => 2 = 1 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 2 = 1 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
 => ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
 => ? = 1 + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 1 + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
 => ? = 2 + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[[[[.,[[.,.],.]],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
 => ([(0,10),(1,9),(2,10),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],11)
 => ? = 1 + 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
 => ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
 => ? = 1 + 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,4),(6,3),(8,5)],9)
 => ? = 2 + 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 1 + 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
 => ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
 => ? = 2 + 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [[[[.,[[[.,.],.],.]],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
 => ([(0,8),(1,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,7)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 0 + 1
[]
 => []
 => .
 => ?
 => ? = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St000527
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[.,.],.],.],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[[[[.,[.,.]],.],.],.],[.,.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => 2 = 1 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[[.,[[.,.],.]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[[.,.],[.,.]],.],.]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 2 = 1 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[[[.,[.,.]],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[[[[[.,.],.],.],.],.],.],.]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1 = 0 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[.,[[.,.],.]],.],.],[.,.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => 2 = 1 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 2 = 1 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
 => ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
 => ? = 1 + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 1 + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
 => ? = 2 + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[[[[.,[[.,.],.]],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [.,[[[[[[.,.],[.,.]],.],.],.],.]]
 => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
 => ? = 1 + 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
 => ([(0,10),(1,9),(2,10),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],11)
 => ? = 1 + 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
 => ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
 => ? = 1 + 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,4),(6,3),(8,5)],9)
 => ? = 2 + 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 1 + 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
 => ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
 => ? = 2 + 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [[[[.,[[[.,.],.],.]],.],.],[.,.]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 1 + 1
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [.,[[[[[[.,.],.],[.,.]],.],.],.]]
 => ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
 => ? = 1 + 1
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
 => ([(0,8),(1,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,7)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 0 + 1
[]
 => []
 => .
 => ?
 => ? = 0 + 1
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000647The number of big descents of a permutation. St000568The hook number of a binary tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000632The jump number of the poset. St001840The number of descents of a set partition. St000353The number of inner valleys of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000523The number of 2-protected nodes of a rooted tree. St000779The tier of a permutation. St000251The number of nonsingleton blocks of a set partition. St000023The number of inner peaks of a permutation. St000092The number of outer peaks of a permutation. St000360The number of occurrences of the pattern 32-1. St000646The number of big ascents of a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000099The number of valleys of a permutation, including the boundary. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000252The number of nodes of degree 3 of a binary tree. St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001487The number of inner corners of a skew partition. St001569The maximal modular displacement of a permutation.