Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000377
Mapping
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 1
[3]
 => 1
[2,1]
 => 0
[1,1,1]
 => 2
[4]
 => 2
[3,1]
 => 0
[2,2]
 => 1
[2,1,1]
 => 2
[1,1,1,1]
 => 3
[5]
 => 3
[4,1]
 => 2
[3,2]
 => 0
[3,1,1]
 => 1
[2,2,1]
 => 2
[2,1,1,1]
 => 3
[1,1,1,1,1]
 => 4
[6]
 => 4
[5,1]
 => 3
[4,2]
 => 1
[4,1,1]
 => 2
[3,3]
 => 2
[3,2,1]
 => 0
[3,1,1,1]
 => 3
[2,2,2]
 => 3
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => 5
[7]
 => 5
[6,1]
 => 4
[5,2]
 => 3
[5,1,1]
 => 4
[4,3]
 => 2
[4,2,1]
 => 0
[4,1,1,1]
 => 3
[3,3,1]
 => 1
[3,2,2]
 => 2
[3,2,1,1]
 => 3
[3,1,1,1,1]
 => 4
[2,2,2,1]
 => 4
[2,2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => 6
[8]
 => 6
[7,1]
 => 5
[6,2]
 => 4
[6,1,1]
 => 5
[5,3]
 => 3
[5,2,1]
 => 3
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001176
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 89%●distinct values known / distinct values provided: 52%
Values
[1]
 => [1]
 => [1]
 => 0
[2]
 => [1,1]
 => [2]
 => 0
[1,1]
 => [2]
 => [1,1]
 => 1
[3]
 => [2,1]
 => [2,1]
 => 1
[2,1]
 => [1,1,1]
 => [3]
 => 0
[1,1,1]
 => [3]
 => [1,1,1]
 => 2
[4]
 => [2,2]
 => [2,2]
 => 2
[3,1]
 => [1,1,1,1]
 => [4]
 => 0
[2,2]
 => [2,1,1]
 => [3,1]
 => 1
[2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[1,1,1,1]
 => [4]
 => [1,1,1,1]
 => 3
[5]
 => [3,2]
 => [2,2,1]
 => 3
[4,1]
 => [3,1,1]
 => [3,1,1]
 => 2
[3,2]
 => [1,1,1,1,1]
 => [5]
 => 0
[3,1,1]
 => [2,1,1,1]
 => [4,1]
 => 1
[2,2,1]
 => [2,2,1]
 => [3,2]
 => 2
[2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,1,1,1,1]
 => 4
[6]
 => [3,3]
 => [2,2,2]
 => 4
[5,1]
 => [3,2,1]
 => [3,2,1]
 => 3
[4,2]
 => [2,1,1,1,1]
 => [5,1]
 => 1
[4,1,1]
 => [2,2,1,1]
 => [4,2]
 => 2
[3,3]
 => [3,1,1,1]
 => [4,1,1]
 => 2
[3,2,1]
 => [1,1,1,1,1,1]
 => [6]
 => 0
[3,1,1,1]
 => [4,1,1]
 => [3,1,1,1]
 => 3
[2,2,2]
 => [2,2,2]
 => [3,3]
 => 3
[2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => 4
[2,1,1,1,1]
 => [5,1]
 => [2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [1,1,1,1,1,1]
 => 5
[7]
 => [4,3]
 => [2,2,2,1]
 => 5
[6,1]
 => [3,3,1]
 => [3,2,2]
 => 4
[5,2]
 => [3,2,1,1]
 => [4,2,1]
 => 3
[5,1,1]
 => [4,2,1]
 => [3,2,1,1]
 => 4
[4,3]
 => [2,2,1,1,1]
 => [5,2]
 => 2
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => 0
[4,1,1,1]
 => [2,2,2,1]
 => [4,3]
 => 3
[3,3,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [5,1,1]
 => 2
[3,2,1,1]
 => [4,1,1,1]
 => [4,1,1,1]
 => 3
[3,1,1,1,1]
 => [5,1,1]
 => [3,1,1,1,1]
 => 4
[2,2,2,1]
 => [3,2,2]
 => [3,3,1]
 => 4
[2,2,1,1,1]
 => [5,2]
 => [2,2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => [6,1]
 => [2,1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => [7]
 => [1,1,1,1,1,1,1]
 => 6
[8]
 => [4,4]
 => [2,2,2,2]
 => 6
[7,1]
 => [4,3,1]
 => [3,2,2,1]
 => 5
[6,2]
 => [3,3,1,1]
 => [4,2,2]
 => 4
[6,1,1]
 => [4,2,2]
 => [3,3,1,1]
 => 5
[5,3]
 => [2,2,2,1,1]
 => [5,3]
 => 3
[5,2,1]
 => [4,1,1,1,1]
 => [5,1,1,1]
 => 3
[]
 => ?
 => ?
 => ? = 0
[6,5,4,3,2,1]
 => ?
 => ?
 => ? = 0
[7,6,5,4,3,2,1]
 => ?
 => ?
 => ? = 0
[7,5,4,3,1]
 => ?
 => ?
 => ? = 2
[11,7,3]
 => ?
 => ?
 => ? = 15
[9,6,3]
 => ?
 => ?
 => ? = 12
[8,6,4,2]
 => ?
 => ?
 => ? = 9
[8,6,4]
 => ?
 => ?
 => ? = 10
[10,6,4]
 => ?
 => ?
 => ? = 13
[11,7,5,1]
 => ?
 => ?
 => ? = 16
[9,7,5,3,1]
 => ?
 => ?
 => ? = 12
[11,9,7,5,3,1]
 => ?
 => ?
 => ? = 20
[9,7,5,5,3,1]
 => ?
 => ?
 => ? = 10
[11,8,7,5,4,1]
 => ?
 => ?
 => ? = 19
[8,7,6,5,4,3,2,1]
 => ?
 => ?
 => ? = 0
[9,6,4]
 => ?
 => ?
 => ? = 12
[10,7,3]
 => ?
 => ?
 => ? = 14
[8,5,4,2]
 => ?
 => ?
 => ? = 8
[9,6,5,3]
 => ?
 => ?
 => ? = 12
[10,7,5,3]
 => ?
 => ?
 => ? = 15
[9,6,4,3]
 => ?
 => ?
 => ? = 11
[10,7,6,4,1]
 => ?
 => ?
 => ? = 16
[11,8,6,5,1]
 => ?
 => ?
 => ? = 19
[11,8,6,4,1]
 => ?
 => ?
 => ? = 19
[12,9,7,5,1]
 => ?
 => ?
 => ? = 23
[13,9,7,5,1]
 => ?
 => ?
 => ? = 24
[9,7,4]
 => ?
 => ?
 => ? = 13
[9,7,5,4,1]
 => ?
 => ?
 => ? = 13
[10,8,6,4,1]
 => ?
 => ?
 => ? = 17
[11,9,7,5,5,3]
 => ?
 => ?
 => ? = 20
[8,6,5,3,1]
 => ?
 => ?
 => ? = 8
[9,7,6,4,2]
 => ?
 => ?
 => ? = 12
[10,8,5,4,1]
 => ?
 => ?
 => ? = 17
[10,8,6,4,2]
 => ?
 => ?
 => ? = 16
[11,9,7,7,5,3,3]
 => ?
 => ?
 => ? = 18
[8,5,5,1]
 => ?
 => ?
 => ? = 11
[9,5,5,1]
 => ?
 => ?
 => ? = 13
[9,7,5,3]
 => ?
 => ?
 => ? = 13
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 89%●distinct values known / distinct values provided: 48%
Values
[1]
 => [1]
 => [1]
 => []
 => 0
[2]
 => [1,1]
 => [2]
 => []
 => 0
[1,1]
 => [2]
 => [1,1]
 => [1]
 => 1
[3]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[2,1]
 => [1,1,1]
 => [3]
 => []
 => 0
[1,1,1]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[4]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[3,1]
 => [1,1,1,1]
 => [4]
 => []
 => 0
[2,2]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[2,1,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
[1,1,1,1]
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3
[5]
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
[4,1]
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
[3,2]
 => [1,1,1,1,1]
 => [5]
 => []
 => 0
[3,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
[2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
[6]
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 4
[5,1]
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 3
[4,2]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 1
[4,1,1]
 => [2,2,1,1]
 => [4,2]
 => [2]
 => 2
[3,3]
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 2
[3,2,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => 0
[3,1,1,1]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 3
[2,2,2]
 => [2,2,2]
 => [3,3]
 => [3]
 => 3
[2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 4
[2,1,1,1,1]
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 5
[7]
 => [4,3]
 => [2,2,2,1]
 => [2,2,1]
 => 5
[6,1]
 => [3,3,1]
 => [3,2,2]
 => [2,2]
 => 4
[5,2]
 => [3,2,1,1]
 => [4,2,1]
 => [2,1]
 => 3
[5,1,1]
 => [4,2,1]
 => [3,2,1,1]
 => [2,1,1]
 => 4
[4,3]
 => [2,2,1,1,1]
 => [5,2]
 => [2]
 => 2
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => []
 => 0
[4,1,1,1]
 => [2,2,2,1]
 => [4,3]
 => [3]
 => 3
[3,3,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => [1]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 2
[3,2,1,1]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,1,1]
 => 3
[3,1,1,1,1]
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 4
[2,2,2,1]
 => [3,2,2]
 => [3,3,1]
 => [3,1]
 => 4
[2,2,1,1,1]
 => [5,2]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 6
[8]
 => [4,4]
 => [2,2,2,2]
 => [2,2,2]
 => 6
[7,1]
 => [4,3,1]
 => [3,2,2,1]
 => [2,2,1]
 => 5
[6,2]
 => [3,3,1,1]
 => [4,2,2]
 => [2,2]
 => 4
[6,1,1]
 => [4,2,2]
 => [3,3,1,1]
 => [3,1,1]
 => 5
[5,3]
 => [2,2,2,1,1]
 => [5,3]
 => [3]
 => 3
[5,2,1]
 => [4,1,1,1,1]
 => [5,1,1,1]
 => [1,1,1]
 => 3
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 11
[]
 => ?
 => ?
 => ?
 => ? = 0
[6,5,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 0
[7,6,5,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 0
[7,5,4,3,1]
 => ?
 => ?
 => ?
 => ? = 2
[11,7,3]
 => ?
 => ?
 => ?
 => ? = 15
[9,6,3]
 => ?
 => ?
 => ?
 => ? = 12
[8,6,4,2]
 => ?
 => ?
 => ?
 => ? = 9
[8,6,4]
 => ?
 => ?
 => ?
 => ? = 10
[10,6,4]
 => ?
 => ?
 => ?
 => ? = 13
[11,7,5,1]
 => ?
 => ?
 => ?
 => ? = 16
[9,7,5,3,1]
 => ?
 => ?
 => ?
 => ? = 12
[11,9,7,5,3,1]
 => ?
 => ?
 => ?
 => ? = 20
[9,7,5,5,3,1]
 => ?
 => ?
 => ?
 => ? = 10
[11,8,7,5,4,1]
 => ?
 => ?
 => ?
 => ? = 19
[8,7,6,5,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 0
[9,6,4]
 => ?
 => ?
 => ?
 => ? = 12
[10,7,3]
 => ?
 => ?
 => ?
 => ? = 14
[8,5,4,2]
 => ?
 => ?
 => ?
 => ? = 8
[9,6,5,3]
 => ?
 => ?
 => ?
 => ? = 12
[10,7,5,3]
 => ?
 => ?
 => ?
 => ? = 15
[9,6,4,3]
 => ?
 => ?
 => ?
 => ? = 11
[10,7,6,4,1]
 => ?
 => ?
 => ?
 => ? = 16
[11,8,6,5,1]
 => ?
 => ?
 => ?
 => ? = 19
[11,8,6,4,1]
 => ?
 => ?
 => ?
 => ? = 19
[12,9,7,5,1]
 => ?
 => ?
 => ?
 => ? = 23
[13,9,7,5,1]
 => ?
 => ?
 => ?
 => ? = 24
[9,7,4]
 => ?
 => ?
 => ?
 => ? = 13
[9,7,5,4,1]
 => ?
 => ?
 => ?
 => ? = 13
[10,8,6,4,1]
 => ?
 => ?
 => ?
 => ? = 17
[11,9,7,5,5,3]
 => ?
 => ?
 => ?
 => ? = 20
[8,6,5,3,1]
 => ?
 => ?
 => ?
 => ? = 8
[9,7,6,4,2]
 => ?
 => ?
 => ?
 => ? = 12
[10,8,5,4,1]
 => ?
 => ?
 => ?
 => ? = 17
[10,8,6,4,2]
 => ?
 => ?
 => ?
 => ? = 16
[11,9,7,7,5,3,3]
 => ?
 => ?
 => ?
 => ? = 18
[8,5,5,1]
 => ?
 => ?
 => ?
 => ? = 11
[9,5,5,1]
 => ?
 => ?
 => ?
 => ? = 13
[9,7,5,3]
 => ?
 => ?
 => ?
 => ? = 13
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000376
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 69%●distinct values known / distinct values provided: 48%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 3
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 4
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 5
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => 4
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 3
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 4
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 4
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 5
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 6
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 5
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => 4
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => 5
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 3
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 4
[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]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 7
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[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]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 7
[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]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 8
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 5
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[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]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 7
[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]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[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]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[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]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 9
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 9
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 8
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 8
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 8
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 5
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 6
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 6
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 7
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 7
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 9
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 9
[2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 10
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 10
[11,1]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 9
[10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 8
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 9
[9,3]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 8
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[9,1,1,1]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 9
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 8
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[8,2,2]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 8
[8,1,1,1,1]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 9
[7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
 => ? = 5
[7,3,1,1]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => ? = 6
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 9
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 6
Description
The bounce deficit of a Dyck path. For a Dyck path $D$ of semilength $n$, this is defined as $$\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).$$ The zeta map [[Mp00032]] sends this statistic to the dinv deficit [[St000369]], both are thus equidistributed.
Matching statistic: St000369
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 65%●distinct values known / distinct values provided: 48%
Values
[1]
 => [1,0,1,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => 0
[1,1]
 => [1,0,1,1,0,0]
 => 1
[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]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 3
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 4
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 4
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[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]
 => 3
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 4
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 4
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 5
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 5
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 4
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 3
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 4
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[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]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 5
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 6
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 6
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 5
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => 4
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 5
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 3
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 4
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[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]
 => ? = 7
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 7
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 5
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? = 6
[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]
 => ? = 7
[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]
 => ? = 8
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 8
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 7
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => ? = 6
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 6
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 5
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 7
[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]
 => ? = 7
[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]
 => ? = 8
[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]
 => ? = 8
[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]
 => ? = 9
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 9
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 8
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 8
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => ? = 7
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 6
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 8
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 7
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 5
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => ? = 6
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 7
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 8
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? = 8
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 7
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? = 8
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 9
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? = 9
[2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 10
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 10
[11,1]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 9
[10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 8
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 9
[9,3]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
 => ? = 8
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[9,1,1,1]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 9
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => ? = 8
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 6
[8,2,2]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
 => ? = 7
Description
The dinv deficit of a Dyck path. For a Dyck path $D$ of semilength $n$, this is defined as $$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$ In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$. See also [[St000376]] for the bounce deficit.
Matching statistic: St000394
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 56%●distinct values known / distinct values provided: 48%
Values
[1]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[2]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[3]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[4]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[3,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[2,2]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[5]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[4,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[3,2]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,1,1,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[6]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[5,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[4,2]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[4,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[3,3]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 2
[3,2,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3
[2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[2,1,1,1,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5
[7]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5
[6,1]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[5,2]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 3
[5,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 4
[4,3]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 2
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[4,1,1,1]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[3,3,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => 2
[3,2,1,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => 3
[3,1,1,1,1]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => 4
[2,2,2,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4
[2,2,1,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 5
[2,1,1,1,1,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 5
[1,1,1,1,1,1,1]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 6
[8]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6
[7,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 5
[6,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 4
[6,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 5
[5,3]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 3
[5,2,1]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => 3
[8,2,1]
 => [4,4,1,1,1]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 6
[7,3,1]
 => [4,3,1,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[7,2,2]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 6
[7,2,1,1]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 7
[6,3,2]
 => [4,2,1,1,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[6,3,1,1]
 => [5,2,1,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[6,2,2,1]
 => [5,2,2,1,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 6
[5,5,1]
 => [5,3,1,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 6
[5,3,3]
 => [3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3
[5,3,1,1,1]
 => [3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[5,2,2,1,1]
 => [4,2,2,1,1,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 5
[5,2,1,1,1,1]
 => [6,3,1,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0]
 => ? = 7
[4,4,1,1,1]
 => [3,3,2,1,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 5
[4,3,3,1]
 => [3,1,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2
[4,3,2,2]
 => [4,1,1,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3
[4,3,2,1,1]
 => [5,1,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[4,3,1,1,1,1]
 => [6,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5
[4,2,2,1,1,1]
 => [6,2,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 6
[4,2,1,1,1,1,1]
 => [7,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 6
[4,1,1,1,1,1,1,1]
 => [8,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
 => ? = 8
[3,3,2,1,1,1]
 => [6,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0,1,0]
 => ? = 7
[3,3,1,1,1,1,1]
 => [7,2,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,1,0]
 => ? = 7
[3,2,2,1,1,1,1]
 => [7,3,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
 => ? = 8
[3,2,1,1,1,1,1,1]
 => [8,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 7
[3,1,1,1,1,1,1,1,1]
 => [9,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 8
[10,2]
 => [5,5,1,1]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 8
[9,2,1]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 7
[8,3,1]
 => [4,4,1,1,1,1]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 6
[8,2,2]
 => [4,4,2,1,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 7
[8,2,1,1]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => ? = 8
[7,4,1]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 7
[7,3,2]
 => [4,3,1,1,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5
[7,3,1,1]
 => [3,3,2,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 6
[7,2,2,1]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 7
[7,2,1,1,1]
 => [6,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ?
 => ? = 8
[6,6]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 8
[6,4,2]
 => [3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[6,4,1,1]
 => [3,2,2,2,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 5
[6,3,3]
 => [3,3,2,1,1,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[6,3,2,1]
 => [5,1,1,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 4
[6,3,1,1,1]
 => [4,3,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
 => ?
 => ? = 6
[6,2,2,2]
 => [4,2,2,2,1,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6
[6,2,2,1,1]
 => [4,3,2,2,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 7
[6,1,1,1,1,1,1]
 => [7,3,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 9
[5,5,2]
 => [5,2,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 5
[5,5,1,1]
 => [5,2,2,1,1,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6
[5,4,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 0
[5,3,3,1]
 => [2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[5,3,2,2]
 => [3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 2
[5,3,2,1,1]
 => [4,1,1,1,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 3
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 52%
Values
[1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[3]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[2,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[1,1,1]
 => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
[4]
 => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[3,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[2,2]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 1 + 1
[2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 3 = 2 + 1
[1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[5]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 4 = 3 + 1
[4,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[3,2]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[3,1,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 2 = 1 + 1
[2,2,1]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 3 = 2 + 1
[2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4 = 3 + 1
[1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
[6]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 5 = 4 + 1
[5,1]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 4 = 3 + 1
[4,2]
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 2 = 1 + 1
[4,1,1]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3 = 2 + 1
[3,3]
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 3 = 2 + 1
[3,2,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 1 = 0 + 1
[3,1,1,1]
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => 4 = 3 + 1
[2,2,2]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 4 = 3 + 1
[2,2,1,1]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 5 = 4 + 1
[2,1,1,1,1]
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => 5 = 4 + 1
[1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => 6 = 5 + 1
[7]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 6 = 5 + 1
[6,1]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 5 = 4 + 1
[5,2]
 => [3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => 4 = 3 + 1
[5,1,1]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => 5 = 4 + 1
[4,3]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 1 = 0 + 1
[4,1,1,1]
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 4 = 3 + 1
[3,3,1]
 => [2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 2 = 1 + 1
[3,2,2]
 => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => 3 = 2 + 1
[3,2,1,1]
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => 5 = 4 + 1
[2,2,2,1]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 5 = 4 + 1
[2,2,1,1,1]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 6 = 5 + 1
[2,1,1,1,1,1]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => 6 = 5 + 1
[1,1,1,1,1,1,1]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 7 = 6 + 1
[8]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 7 = 6 + 1
[7,1]
 => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => 6 = 5 + 1
[6,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 5 = 4 + 1
[6,1,1]
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 6 = 5 + 1
[5,3]
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => 4 = 3 + 1
[5,2,1]
 => [4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => 4 = 3 + 1
[11]
 => [6,5]
 => [[1,2,3,4,5,11],[6,7,8,9,10]]
 => ? = 9 + 1
[10,1]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? = 8 + 1
[7,3,1]
 => [4,3,1,1,1,1]
 => [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
 => ? = 5 + 1
[7,1,1,1,1]
 => [6,3,2]
 => [[1,2,5,9,10,11],[3,4,8],[6,7]]
 => ? = 8 + 1
[6,5]
 => [3,2,2,2,2]
 => [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
 => ? = 6 + 1
[6,4,1]
 => [3,2,2,2,1,1]
 => [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
 => ? = 5 + 1
[6,3,2]
 => [4,2,1,1,1,1,1]
 => [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
 => ? = 4 + 1
[6,3,1,1]
 => [5,2,1,1,1,1]
 => [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
 => ? = 5 + 1
[5,4,2]
 => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ? = 2 + 1
[5,4,1,1]
 => [2,2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
 => ? = 3 + 1
[5,3,3]
 => [3,2,1,1,1,1,1,1]
 => [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
 => ? = 3 + 1
[5,3,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 0 + 1
[5,3,1,1,1]
 => [3,2,2,1,1,1,1]
 => [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
 => ? = 4 + 1
[5,2,2,2]
 => [2,2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
 => ? = 4 + 1
[5,2,2,1,1]
 => [4,2,2,1,1,1]
 => [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
 => ? = 5 + 1
[5,2,1,1,1,1]
 => [6,3,1,1]
 => [[1,4,5,9,10,11],[2,7,8],[3],[6]]
 => ? = 7 + 1
[5,1,1,1,1,1,1]
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? = 8 + 1
[4,4,3]
 => [3,3,1,1,1,1,1]
 => [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
 => ? = 4 + 1
[4,4,2,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 1 + 1
[4,4,1,1,1]
 => [3,3,2,1,1,1]
 => [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
 => ? = 5 + 1
[4,3,3,1]
 => [3,1,1,1,1,1,1,1,1]
 => [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? = 2 + 1
[4,3,2,2]
 => [4,1,1,1,1,1,1,1]
 => [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
 => ? = 3 + 1
[4,3,2,1,1]
 => [5,1,1,1,1,1,1]
 => [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
 => ? = 4 + 1
[4,3,1,1,1,1]
 => [6,1,1,1,1,1]
 => [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
 => ? = 5 + 1
[4,2,2,1,1,1]
 => [6,2,1,1,1]
 => [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
 => ? = 6 + 1
[4,2,1,1,1,1,1]
 => [7,1,1,1,1]
 => [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
 => ? = 6 + 1
[4,1,1,1,1,1,1,1]
 => [8,2,1]
 => [[1,3,6,7,8,9,10,11],[2,5],[4]]
 => ? = 8 + 1
[3,3,3,2]
 => [2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? = 5 + 1
[3,3,2,1,1,1]
 => [6,2,2,1]
 => [[1,3,8,9,10,11],[2,5],[4,7],[6]]
 => ? = 7 + 1
[3,3,1,1,1,1,1]
 => [7,2,1,1]
 => [[1,4,7,8,9,10,11],[2,6],[3],[5]]
 => ? = 7 + 1
[3,2,2,2,1,1]
 => [6,4,1]
 => [[1,3,4,5,10,11],[2,7,8,9],[6]]
 => ? = 8 + 1
[3,2,2,1,1,1,1]
 => [7,3,1]
 => [[1,3,4,8,9,10,11],[2,6,7],[5]]
 => ? = 8 + 1
[3,2,1,1,1,1,1,1]
 => [8,1,1,1]
 => [[1,5,6,7,8,9,10,11],[2],[3],[4]]
 => ? = 7 + 1
[3,1,1,1,1,1,1,1,1]
 => [9,1,1]
 => [[1,4,5,6,7,8,9,10,11],[2],[3]]
 => ? = 8 + 1
[2,2,2,2,1,1,1]
 => [7,4]
 => [[1,2,3,4,9,10,11],[5,6,7,8]]
 => ? = 9 + 1
[2,2,2,1,1,1,1,1]
 => [8,3]
 => [[1,2,3,7,8,9,10,11],[4,5,6]]
 => ? = 9 + 1
[2,2,1,1,1,1,1,1,1]
 => [9,2]
 => [[1,2,5,6,7,8,9,10,11],[3,4]]
 => ? = 9 + 1
[2,1,1,1,1,1,1,1,1,1]
 => [10,1]
 => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 9 + 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [11]
 => [[1,2,3,4,5,6,7,8,9,10,11]]
 => ? = 10 + 1
[11,1]
 => [6,5,1]
 => [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
 => ? = 9 + 1
[10,2]
 => [5,5,1,1]
 => [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
 => ? = 8 + 1
[10,1,1]
 => [5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ? = 9 + 1
[8,3,1]
 => [4,4,1,1,1,1]
 => [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
 => ? = 6 + 1
[8,1,1,1,1]
 => [6,3,3]
 => [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
 => ? = 9 + 1
[7,5]
 => [3,3,2,2,2]
 => [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
 => ? = 7 + 1
[7,3,2]
 => [4,3,1,1,1,1,1]
 => [[1,7,8,12],[2,10,11],[3],[4],[5],[6],[9]]
 => ? = 5 + 1
[7,2,1,1,1]
 => [6,3,2,1]
 => [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
 => ? = 8 + 1
[7,1,1,1,1,1]
 => [3,3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
 => ? = 8 + 1
[6,5,1]
 => [3,2,2,2,2,1]
 => [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? = 6 + 1
[6,4,2]
 => [3,2,2,1,1,1,1,1]
 => [[1,7,12],[2,9],[3,11],[4],[5],[6],[8],[10]]
 => ? = 4 + 1
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000738
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 48% ●values known / values provided: 50%●distinct values known / distinct values provided: 48%
Values
[1]
 => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[2]
 => [1,1]
 => [2]
 => [[1,2]]
 => 1 = 0 + 1
[1,1]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[3]
 => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[2,1]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[1,1,1]
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[4]
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[3,1]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[2,2]
 => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[2,1,1]
 => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,1,1,1]
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[5]
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4 = 3 + 1
[4,1]
 => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[3,2]
 => [1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[3,1,1]
 => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[2,2,1]
 => [2,2,1]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 3 = 2 + 1
[2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,1,1,1,1]
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5 = 4 + 1
[6]
 => [3,3]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 5 = 4 + 1
[5,1]
 => [3,2,1]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 4 = 3 + 1
[4,2]
 => [2,1,1,1,1]
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => 2 = 1 + 1
[4,1,1]
 => [2,2,1,1]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 3 = 2 + 1
[3,3]
 => [3,1,1,1]
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => 3 = 2 + 1
[3,2,1]
 => [1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => 1 = 0 + 1
[3,1,1,1]
 => [4,1,1]
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 4 = 3 + 1
[2,2,2]
 => [2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 4 = 3 + 1
[2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 5 = 4 + 1
[2,1,1,1,1]
 => [5,1]
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 5 = 4 + 1
[1,1,1,1,1,1]
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
[7]
 => [4,3]
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 6 = 5 + 1
[6,1]
 => [3,3,1]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 5 = 4 + 1
[5,2]
 => [3,2,1,1]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => 4 = 3 + 1
[5,1,1]
 => [4,2,1]
 => [3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => 5 = 4 + 1
[4,3]
 => [2,2,1,1,1]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 1 = 0 + 1
[4,1,1,1]
 => [2,2,2,1]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 4 = 3 + 1
[3,3,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => 2 = 1 + 1
[3,2,2]
 => [3,1,1,1,1]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => 3 = 2 + 1
[3,2,1,1]
 => [4,1,1,1]
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [5,1,1]
 => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => 5 = 4 + 1
[2,2,2,1]
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 5 = 4 + 1
[2,2,1,1,1]
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 6 = 5 + 1
[2,1,1,1,1,1]
 => [6,1]
 => [2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
[1,1,1,1,1,1,1]
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 1
[8]
 => [4,4]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 7 = 6 + 1
[7,1]
 => [4,3,1]
 => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => 6 = 5 + 1
[6,2]
 => [3,3,1,1]
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 5 = 4 + 1
[6,1,1]
 => [4,2,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 6 = 5 + 1
[5,3]
 => [2,2,2,1,1]
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => 4 = 3 + 1
[5,2,1]
 => [4,1,1,1,1]
 => [5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => 4 = 3 + 1
[11]
 => [6,5]
 => [2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? = 9 + 1
[10,1]
 => [5,5,1]
 => [3,2,2,2,2]
 => [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
 => ? = 8 + 1
[7,3,1]
 => [4,3,1,1,1,1]
 => [6,2,2,1]
 => [[1,3,8,9,10,11],[2,5],[4,7],[6]]
 => ? = 5 + 1
[7,1,1,1,1]
 => [6,3,2]
 => [3,3,2,1,1,1]
 => [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
 => ? = 8 + 1
[6,5]
 => [3,2,2,2,2]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? = 6 + 1
[6,4,1]
 => [3,2,2,2,1,1]
 => [6,4,1]
 => [[1,3,4,5,10,11],[2,7,8,9],[6]]
 => ? = 5 + 1
[6,3,2]
 => [4,2,1,1,1,1,1]
 => [7,2,1,1]
 => [[1,4,7,8,9,10,11],[2,6],[3],[5]]
 => ? = 4 + 1
[6,3,1,1]
 => [5,2,1,1,1,1]
 => [6,2,1,1,1]
 => [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
 => ? = 5 + 1
[5,4,2]
 => [2,2,1,1,1,1,1,1,1]
 => [9,2]
 => [[1,2,5,6,7,8,9,10,11],[3,4]]
 => ? = 2 + 1
[5,4,1,1]
 => [2,2,2,1,1,1,1,1]
 => [8,3]
 => [[1,2,3,7,8,9,10,11],[4,5,6]]
 => ? = 3 + 1
[5,3,3]
 => [3,2,1,1,1,1,1,1]
 => [8,2,1]
 => [[1,3,6,7,8,9,10,11],[2,5],[4]]
 => ? = 3 + 1
[5,3,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [11]
 => [[1,2,3,4,5,6,7,8,9,10,11]]
 => ? = 0 + 1
[5,3,1,1,1]
 => [3,2,2,1,1,1,1]
 => [7,3,1]
 => [[1,3,4,8,9,10,11],[2,6,7],[5]]
 => ? = 4 + 1
[5,2,2,2]
 => [2,2,2,2,1,1,1]
 => [7,4]
 => [[1,2,3,4,9,10,11],[5,6,7,8]]
 => ? = 4 + 1
[5,2,2,1,1]
 => [4,2,2,1,1,1]
 => [6,3,1,1]
 => [[1,4,5,9,10,11],[2,7,8],[3],[6]]
 => ? = 5 + 1
[5,2,1,1,1,1]
 => [6,3,1,1]
 => [4,2,2,1,1,1]
 => [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
 => ? = 7 + 1
[5,1,1,1,1,1,1]
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
 => ? = 8 + 1
[4,4,3]
 => [3,3,1,1,1,1,1]
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? = 4 + 1
[4,4,2,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [10,1]
 => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1 + 1
[4,4,1,1,1]
 => [3,3,2,1,1,1]
 => [6,3,2]
 => [[1,2,5,9,10,11],[3,4,8],[6,7]]
 => ? = 5 + 1
[4,3,3,1]
 => [3,1,1,1,1,1,1,1,1]
 => [9,1,1]
 => [[1,4,5,6,7,8,9,10,11],[2],[3]]
 => ? = 2 + 1
[4,3,2,2]
 => [4,1,1,1,1,1,1,1]
 => [8,1,1,1]
 => [[1,5,6,7,8,9,10,11],[2],[3],[4]]
 => ? = 3 + 1
[4,3,2,1,1]
 => [5,1,1,1,1,1,1]
 => [7,1,1,1,1]
 => [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
 => ? = 4 + 1
[4,3,1,1,1,1]
 => [6,1,1,1,1,1]
 => [6,1,1,1,1,1]
 => [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
 => ? = 5 + 1
[4,2,2,1,1,1]
 => [6,2,1,1,1]
 => [5,2,1,1,1,1]
 => [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
 => ? = 6 + 1
[4,2,1,1,1,1,1]
 => [7,1,1,1,1]
 => [5,1,1,1,1,1,1]
 => [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
 => ? = 6 + 1
[4,1,1,1,1,1,1,1]
 => [8,2,1]
 => [3,2,1,1,1,1,1,1]
 => [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
 => ? = 8 + 1
[3,3,3,2]
 => [2,2,2,2,2,1]
 => [6,5]
 => [[1,2,3,4,5,11],[6,7,8,9,10]]
 => ? = 5 + 1
[3,3,2,1,1,1]
 => [6,2,2,1]
 => [4,3,1,1,1,1]
 => [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
 => ? = 7 + 1
[3,3,1,1,1,1,1]
 => [7,2,1,1]
 => [4,2,1,1,1,1,1]
 => [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
 => ? = 7 + 1
[3,2,2,2,1,1]
 => [6,4,1]
 => [3,2,2,2,1,1]
 => [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
 => ? = 8 + 1
[3,2,2,1,1,1,1]
 => [7,3,1]
 => [3,2,2,1,1,1,1]
 => [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
 => ? = 8 + 1
[3,2,1,1,1,1,1,1]
 => [8,1,1,1]
 => [4,1,1,1,1,1,1,1]
 => [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
 => ? = 7 + 1
[3,1,1,1,1,1,1,1,1]
 => [9,1,1]
 => [3,1,1,1,1,1,1,1,1]
 => [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? = 8 + 1
[2,2,2,2,1,1,1]
 => [7,4]
 => [2,2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
 => ? = 9 + 1
[2,2,2,1,1,1,1,1]
 => [8,3]
 => [2,2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
 => ? = 9 + 1
[2,2,1,1,1,1,1,1,1]
 => [9,2]
 => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ? = 9 + 1
[2,1,1,1,1,1,1,1,1,1]
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 9 + 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10 + 1
[11,1]
 => [6,5,1]
 => [3,2,2,2,2,1]
 => [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? = 9 + 1
[10,2]
 => [5,5,1,1]
 => [4,2,2,2,2]
 => [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
 => ? = 8 + 1
[10,1,1]
 => [5,5,2]
 => [3,3,2,2,2]
 => [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
 => ? = 9 + 1
[8,3,1]
 => [4,4,1,1,1,1]
 => [6,2,2,2]
 => [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
 => ? = 6 + 1
[8,1,1,1,1]
 => [6,3,3]
 => [3,3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3,11,12],[4],[7],[10]]
 => ? = 9 + 1
[7,5]
 => [3,3,2,2,2]
 => [5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ? = 7 + 1
[7,3,2]
 => [4,3,1,1,1,1,1]
 => [7,2,2,1]
 => [[1,3,8,9,10,11,12],[2,5],[4,7],[6]]
 => ? = 5 + 1
[7,2,1,1,1]
 => [6,3,2,1]
 => [4,3,2,1,1,1]
 => [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
 => ? = 8 + 1
[7,1,1,1,1,1]
 => [3,3,3,3]
 => [4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
 => ? = 8 + 1
[6,5,1]
 => [3,2,2,2,2,1]
 => [6,5,1]
 => [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
 => ? = 6 + 1
[6,4,2]
 => [3,2,2,1,1,1,1,1]
 => [8,3,1]
 => [[1,3,4,8,9,10,11,12],[2,6,7],[5]]
 => ? = 4 + 1
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St000157
(load all 2 compositions to match this statistic)
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 48%
Values
[1]
 => [1]
 => [[1]]
 => [[1]]
 => 0
[2]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 0
[1,1]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[3]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[2,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 0
[1,1,1]
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 2
[4]
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 0
[2,2]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
[5]
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 3
[4,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 2
[3,2]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0
[3,1,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[2,2,1]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 2
[2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 3
[1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
[6]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 4
[5,1]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 3
[4,2]
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 1
[4,1,1]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 2
[3,3]
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => 2
[3,2,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 0
[3,1,1,1]
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [[1,2,3],[4],[5],[6]]
 => 3
[2,2,2]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 3
[2,2,1,1]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 4
[2,1,1,1,1]
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 5
[7]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => 5
[6,1]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [[1,2,5],[3,6],[4,7]]
 => 4
[5,2]
 => [3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6],[7]]
 => 3
[5,1,1]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [[1,2,4],[3,5],[6],[7]]
 => 4
[4,3]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7]]
 => 2
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => 0
[4,1,1,1]
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => 3
[3,3,1]
 => [2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6],[7]]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6],[7]]
 => 2
[3,2,1,1]
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [[1,2,3,4],[5],[6],[7]]
 => 3
[3,1,1,1,1]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7]]
 => 4
[2,2,2,1]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 4
[2,2,1,1,1]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 5
[2,1,1,1,1,1]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 5
[1,1,1,1,1,1,1]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 6
[8]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8]]
 => 6
[7,1]
 => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [[1,2,5],[3,6],[4,7],[8]]
 => 5
[6,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [[1,2,3,6],[4,7],[5,8]]
 => 4
[6,1,1]
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8]]
 => 5
[5,3]
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [[1,2,3,5,7],[4,6,8]]
 => 3
[5,2,1]
 => [4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6],[7],[8]]
 => 3
[11]
 => [6,5]
 => [[1,2,3,4,5,11],[6,7,8,9,10]]
 => [[1,6],[2,7],[3,8],[4,9],[5,10],[11]]
 => ? = 9
[10,1]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => [[1,2,7],[3,8],[4,9],[5,10],[6,11]]
 => ? = 8
[9,2]
 => [5,4,1,1]
 => [[1,4,5,6,11],[2,8,9,10],[3],[7]]
 => [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
 => ? = 7
[9,1,1]
 => [5,4,2]
 => [[1,2,5,6,11],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
 => ? = 8
[8,3]
 => [4,4,2,1]
 => [[1,3,6,7],[2,5,10,11],[4,9],[8]]
 => [[1,2,4,8],[3,5,9],[6,10],[7,11]]
 => ? = 7
[8,2,1]
 => [4,4,1,1,1]
 => [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
 => [[1,2,3,4,8],[5,9],[6,10],[7,11]]
 => ? = 6
[8,1,1,1]
 => [5,3,3]
 => [[1,2,3,10,11],[4,5,6],[7,8,9]]
 => [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
 => ? = 8
[7,4]
 => [5,2,2,2]
 => [[1,2,9,10,11],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
 => ? = 7
[7,3,1]
 => [4,3,1,1,1,1]
 => [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
 => ?
 => ? = 5
[7,2,2]
 => [4,3,2,1,1]
 => [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
 => [[1,2,3,5,8],[4,6,9],[7,10],[11]]
 => ? = 6
[7,2,1,1]
 => [5,3,2,1]
 => [[1,3,6,10,11],[2,5,9],[4,8],[7]]
 => [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
 => ? = 7
[7,1,1,1,1]
 => [6,3,2]
 => [[1,2,5,9,10,11],[3,4,8],[6,7]]
 => ?
 => ? = 8
[6,5]
 => [3,2,2,2,2]
 => [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
 => [[1,3,5,7,9],[2,4,6,8,10],[11]]
 => ? = 6
[6,4,1]
 => [3,2,2,2,1,1]
 => [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
 => ?
 => ? = 5
[6,3,2]
 => [4,2,1,1,1,1,1]
 => [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
 => ?
 => ? = 4
[6,3,1,1]
 => [5,2,1,1,1,1]
 => [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6,8],[9],[10],[11]]
 => ? = 5
[6,2,2,1]
 => [5,2,2,1,1]
 => [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
 => [[1,2,3,5,7],[4,6,8],[9],[10],[11]]
 => ? = 6
[6,2,1,1,1]
 => [3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11]]
 => ? = 6
[6,1,1,1,1,1]
 => [3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => ? = 7
[5,5,1]
 => [5,3,1,1,1]
 => [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
 => [[1,2,3,4,7],[5,8],[6,9],[10],[11]]
 => ? = 6
[5,4,2]
 => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ?
 => ? = 2
[5,4,1,1]
 => [2,2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
 => [[1,2,3,4,5,6,8,10],[7,9,11]]
 => ? = 3
[5,3,3]
 => [3,2,1,1,1,1,1,1]
 => [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
 => ?
 => ? = 3
[5,3,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => [[1,2,3,4,5,6,7,8,9,10,11]]
 => ? = 0
[5,3,1,1,1]
 => [3,2,2,1,1,1,1]
 => [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
 => ?
 => ? = 4
[5,2,2,2]
 => [2,2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
 => ?
 => ? = 4
[5,2,2,1,1]
 => [4,2,2,1,1,1]
 => [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
 => ?
 => ? = 5
[5,2,1,1,1,1]
 => [6,3,1,1]
 => [[1,4,5,9,10,11],[2,7,8],[3],[6]]
 => ?
 => ? = 7
[5,1,1,1,1,1,1]
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8],[9],[10],[11]]
 => ? = 8
[4,4,3]
 => [3,3,1,1,1,1,1]
 => [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
 => [[1,2,3,4,5,6,9],[7,10],[8,11]]
 => ? = 4
[4,4,2,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 1
[4,4,1,1,1]
 => [3,3,2,1,1,1]
 => [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
 => ?
 => ? = 5
[4,3,3,1]
 => [3,1,1,1,1,1,1,1,1]
 => [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ?
 => ? = 2
[4,3,2,2]
 => [4,1,1,1,1,1,1,1]
 => [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
 => ?
 => ? = 3
[4,3,2,1,1]
 => [5,1,1,1,1,1,1]
 => [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
 => ?
 => ? = 4
[4,3,1,1,1,1]
 => [6,1,1,1,1,1]
 => [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
 => ?
 => ? = 5
[4,2,2,2,1]
 => [3,3,3,1,1]
 => [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
 => [[1,2,3,6,9],[4,7,10],[5,8,11]]
 => ? = 6
[4,2,2,1,1,1]
 => [6,2,1,1,1]
 => [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7],[8],[9],[10],[11]]
 => ? = 6
[4,2,1,1,1,1,1]
 => [7,1,1,1,1]
 => [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
 => ?
 => ? = 6
[4,1,1,1,1,1,1,1]
 => [8,2,1]
 => [[1,3,6,7,8,9,10,11],[2,5],[4]]
 => ?
 => ? = 8
[3,3,3,2]
 => [2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
 => [[1,2,4,6,8,10],[3,5,7,9,11]]
 => ? = 5
[3,3,3,1,1]
 => [4,2,2,2,1]
 => [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
 => [[1,2,4,6,8],[3,5,7,9],[10],[11]]
 => ? = 6
[3,3,2,2,1]
 => [4,3,2,2]
 => [[1,2,7,11],[3,4,10],[5,6],[8,9]]
 => [[1,3,5,8],[2,4,6,9],[7,10],[11]]
 => ? = 7
[3,3,2,1,1,1]
 => [6,2,2,1]
 => [[1,3,8,9,10,11],[2,5],[4,7],[6]]
 => ?
 => ? = 7
[3,3,1,1,1,1,1]
 => [7,2,1,1]
 => [[1,4,7,8,9,10,11],[2,6],[3],[5]]
 => ?
 => ? = 7
[3,2,2,2,2]
 => [4,3,3,1]
 => [[1,3,4,11],[2,6,7],[5,9,10],[8]]
 => [[1,2,5,8],[3,6,9],[4,7,10],[11]]
 => ? = 7
[3,2,2,2,1,1]
 => [6,4,1]
 => [[1,3,4,5,10,11],[2,7,8,9],[6]]
 => ?
 => ? = 8
[3,2,2,1,1,1,1]
 => [7,3,1]
 => [[1,3,4,8,9,10,11],[2,6,7],[5]]
 => ?
 => ? = 8
[3,2,1,1,1,1,1,1]
 => [8,1,1,1]
 => [[1,5,6,7,8,9,10,11],[2],[3],[4]]
 => ?
 => ? = 7
[3,1,1,1,1,1,1,1,1]
 => [9,1,1]
 => [[1,4,5,6,7,8,9,10,11],[2],[3]]
 => ?
 => ? = 8
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.