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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1]
=> 0
[2,1]
=> [1]
=> [1]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> 0
[3,1]
=> [1]
=> [1]
=> 0
[2,2]
=> [2]
=> [1,1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[4,1]
=> [1]
=> [1]
=> 0
[3,2]
=> [2]
=> [1,1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[5,1]
=> [1]
=> [1]
=> 0
[4,2]
=> [2]
=> [1,1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[6,1]
=> [1]
=> [1]
=> 0
[5,2]
=> [2]
=> [1,1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[7,1]
=> [1]
=> [1]
=> 0
[6,2]
=> [2]
=> [1,1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> 3
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,2,2]
=> [2,2]
=> [2,2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[3,3,2]
=> [3,2]
=> [2,2,1]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 2
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
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 97%●distinct values known / distinct values provided: 69%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 97%●distinct values known / distinct values provided: 69%
Values
[1,1]
=> [1]
=> [1]
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> []
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[3,1]
=> [1]
=> [1]
=> []
=> 0
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[4,1]
=> [1]
=> [1]
=> []
=> 0
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[5,1]
=> [1]
=> [1]
=> []
=> 0
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[6,1]
=> [1]
=> [1]
=> []
=> 0
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> 0
[7,1]
=> [1]
=> [1]
=> []
=> 0
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[6,5,3,3,3,2,1]
=> [5,3,3,3,2,1]
=> [6,5,4,1,1]
=> [5,4,1,1]
=> ? = 11
[6,5,4,3,2,2,1]
=> [5,4,3,2,2,1]
=> [6,5,3,2,1]
=> [5,3,2,1]
=> ? = 11
[5,5,4,3,2,2,1]
=> [5,4,3,2,2,1]
=> [6,5,3,2,1]
=> [5,3,2,1]
=> ? = 11
[6,5,5,4,1,1,1]
=> [5,5,4,1,1,1]
=> [6,3,3,3,2]
=> [3,3,3,2]
=> ? = 11
[6,5,4,3,3,2]
=> [5,4,3,3,2]
=> [5,5,4,2,1]
=> [5,4,2,1]
=> ? = 12
[6,5,4,3,2,2]
=> [5,4,3,2,2]
=> [5,5,3,2,1]
=> [5,3,2,1]
=> ? = 11
[7,6,5,2,2,2]
=> [6,5,2,2,2]
=> [5,5,2,2,2,1]
=> [5,2,2,2,1]
=> ? = 12
[4,4,4,4,3,1]
=> [4,4,4,3,1]
=> [5,4,4,3]
=> [4,4,3]
=> ? = 11
[5,5,5,4,2,1]
=> [5,5,4,2,1]
=> [5,4,3,3,2]
=> [4,3,3,2]
=> ? = 12
[7,6,5,3,2,1]
=> [6,5,3,2,1]
=> [5,4,3,2,2,1]
=> [4,3,2,2,1]
=> ? = 12
[6,6,5,3,2,1]
=> [6,5,3,2,1]
=> [5,4,3,2,2,1]
=> [4,3,2,2,1]
=> ? = 12
[5,5,5,3,2,1]
=> [5,5,3,2,1]
=> [5,4,3,2,2]
=> [4,3,2,2]
=> ? = 11
[7,6,4,3,2,1]
=> [6,4,3,2,1]
=> [5,4,3,2,1,1]
=> [4,3,2,1,1]
=> ? = 11
[6,6,4,3,2,1]
=> [6,4,3,2,1]
=> [5,4,3,2,1,1]
=> [4,3,2,1,1]
=> ? = 11
[7,6,5,2,2,1]
=> [6,5,2,2,1]
=> [5,4,2,2,2,1]
=> [4,2,2,2,1]
=> ? = 11
[7,6,5,4,1,1]
=> [6,5,4,1,1]
=> [5,3,3,3,2,1]
=> [3,3,3,2,1]
=> ? = 12
[6,5,5,4,3]
=> [5,5,4,3]
=> [4,4,4,3,2]
=> [4,4,3,2]
=> ? = 13
[5,4,4,4,3]
=> [4,4,4,3]
=> [4,4,4,3]
=> [4,4,3]
=> ? = 11
[7,6,5,4,2]
=> [6,5,4,2]
=> [4,4,3,3,2,1]
=> [4,3,3,2,1]
=> ? = 13
[7,6,5,3,2]
=> [6,5,3,2]
=> [4,4,3,2,2,1]
=> [4,3,2,2,1]
=> ? = 12
[7,6,4,3,2]
=> [6,4,3,2]
=> [4,4,3,2,1,1]
=> [4,3,2,1,1]
=> ? = 11
[7,6,5,4,1]
=> [6,5,4,1]
=> [4,3,3,3,2,1]
=> [3,3,3,2,1]
=> ? = 12
[7,6,5,4]
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [3,3,3,2,1]
=> ? = 12
[6,5,5,4]
=> [5,5,4]
=> [3,3,3,3,2]
=> [3,3,3,2]
=> ? = 11
[7,6,5,3]
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [3,3,2,2,1]
=> ? = 11
[4,4,4,4,4]
=> [4,4,4,4]
=> [4,4,4,4]
=> [4,4,4]
=> ? = 12
[4,4,4,3,3,3]
=> [4,4,3,3,3]
=> [5,5,5,2]
=> [5,5,2]
=> ? = 12
[5,5,5,5]
=> [5,5,5]
=> [3,3,3,3,3]
=> [3,3,3,3]
=> ? = 12
[8,7,6]
=> [7,6]
=> [2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> ? = 11
[9,8,7]
=> [8,7]
=> [2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> ? = 13
[15,15]
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
[7,7,7]
=> [7,7]
=> [2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> ? = 12
[9,7,5,3,1]
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> ? = 12
[9,6,5,3]
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [3,3,2,2,1]
=> ? = 11
[10,7,5,3]
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> ? = 12
[9,7,5,4,1]
=> [7,5,4,1]
=> [4,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> ? = 13
[8,6,5,3,1]
=> [6,5,3,1]
=> [4,3,3,2,2,1]
=> [3,3,2,2,1]
=> ? = 11
[9,7,5,3]
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> ? = 12
[8,8,8]
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? = 14
[16,16]
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 15
[8,7,4,3,2,1]
=> [7,4,3,2,1]
=> [5,4,3,2,1,1,1]
=> [4,3,2,1,1,1]
=> ? = 12
[5,4,3,3,3,3,1]
=> [4,3,3,3,3,1]
=> [6,5,5,1]
=> [5,5,1]
=> ? = 11
[6,5,5,5]
=> [5,5,5]
=> [3,3,3,3,3]
=> [3,3,3,3]
=> ? = 12
[5,4,4,4,4]
=> [4,4,4,4]
=> [4,4,4,4]
=> [4,4,4]
=> ? = 12
[8,7,6,4]
=> [7,6,4]
=> [3,3,3,3,2,2,1]
=> [3,3,3,2,2,1]
=> ? = 14
[4,4,4,4,4,1]
=> [4,4,4,4,1]
=> [5,4,4,4]
=> [4,4,4]
=> ? = 12
[5,5,5,5,1]
=> [5,5,5,1]
=> [4,3,3,3,3]
=> [3,3,3,3]
=> ? = 12
[7,7,4,3,2,1]
=> [7,4,3,2,1]
=> [5,4,3,2,1,1,1]
=> [4,3,2,1,1,1]
=> ? = 12
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000377
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 86%●distinct values known / distinct values provided: 69%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 86%●distinct values known / distinct values provided: 69%
Values
[1,1]
=> [1]
=> [1]
=> 0
[2,1]
=> [1]
=> [1]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> 0
[3,1]
=> [1]
=> [1]
=> 0
[2,2]
=> [2]
=> [1,1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[4,1]
=> [1]
=> [1]
=> 0
[3,2]
=> [2]
=> [1,1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,1]
=> [2,1]
=> [3]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[5,1]
=> [1]
=> [1]
=> 0
[4,2]
=> [2]
=> [1,1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> 2
[3,2,1]
=> [2,1]
=> [3]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[2,2,2]
=> [2,2]
=> [4]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[6,1]
=> [1]
=> [1]
=> 0
[5,2]
=> [2]
=> [1,1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> 2
[4,2,1]
=> [2,1]
=> [3]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,2,2]
=> [2,2]
=> [4]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 0
[7,1]
=> [1]
=> [1]
=> 0
[6,2]
=> [2]
=> [1,1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> 2
[5,2,1]
=> [2,1]
=> [3]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> 3
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,2,2]
=> [2,2]
=> [4]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[3,3,2]
=> [3,2]
=> [5]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> ? = 6
[2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? = 5
[2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> [6,4,3]
=> ? = 4
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> ? = 8
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [7,6]
=> ? = 8
[3,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,3,1,1,1,1]
=> ? = 7
[3,3,3,2,2,1,1,1]
=> [3,3,2,2,1,1,1]
=> [6,5,2]
=> ? = 6
[3,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> ? = 6
[3,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? = 5
[3,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> [6,4,3]
=> ? = 4
[2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [6,2,2,2,2]
=> ? = 7
[2,2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,2,1,1]
=> [6,5,1,1,1]
=> ? = 6
[2,2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1]
=> [6,5,3]
=> ? = 5
[4,4,4,4,1]
=> [4,4,4,1]
=> [3,2,2,2,2,2]
=> ? = 9
[4,4,4,3,2]
=> [4,4,3,2]
=> [3,3,2,2,2,1]
=> ? = 9
[4,4,4,3,1,1]
=> [4,4,3,1,1]
=> [4,2,2,2,2,1]
=> ? = 8
[4,4,4,2,2,1]
=> [4,4,2,2,1]
=> [6,2,2,1,1,1]
=> ? = 8
[4,4,4,2,1,1,1]
=> [4,4,2,1,1,1]
=> [8,3,1,1]
=> ? = 7
[4,4,4,1,1,1,1,1]
=> [4,4,1,1,1,1,1]
=> [8,3,2]
=> ? = 6
[4,4,3,3,3]
=> [4,3,3,3]
=> [7,1,1,1,1,1,1]
=> ? = 9
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 8
[4,4,3,3,1,1,1]
=> [4,3,3,1,1,1]
=> [4,3,2,2,2]
=> ? = 7
[4,4,3,2,2,1,1]
=> [4,3,2,2,1,1]
=> [6,5,1,1]
=> ? = 7
[4,4,3,2,1,1,1,1]
=> [4,3,2,1,1,1,1]
=> [6,3,2,1,1]
=> ? = 6
[4,4,3,1,1,1,1,1,1]
=> [4,3,1,1,1,1,1,1]
=> [6,3,2,2]
=> ? = 5
[4,4,2,2,2,2,1]
=> [4,2,2,2,2,1]
=> [5,2,2,2,2]
=> ? = 7
[4,4,2,2,2,1,1,1]
=> [4,2,2,2,1,1,1]
=> [5,5,1,1,1]
=> ? = 6
[4,4,2,2,1,1,1,1,1]
=> [4,2,2,1,1,1,1,1]
=> [7,3,2,1]
=> ? = 5
[4,4,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> [6,3,3,1]
=> ? = 4
[4,4,1,1,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> [6,4,2,1]
=> ? = 3
[4,3,3,3,3,1]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> ? = 8
[4,3,3,3,2,2]
=> [3,3,3,2,2]
=> [7,6]
=> ? = 8
[4,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,3,1,1,1,1]
=> ? = 7
[4,3,3,2,2,1,1,1]
=> [3,3,2,2,1,1,1]
=> [6,5,2]
=> ? = 6
[4,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> ? = 6
[4,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? = 5
[4,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> [6,4,3]
=> ? = 4
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [8,6]
=> ? = 9
[3,3,3,3,3,1,1]
=> [3,3,3,3,1,1]
=> [7,3,1,1,1,1]
=> ? = 8
[3,3,3,3,2,2,1]
=> [3,3,3,2,2,1]
=> [7,6,1]
=> ? = 8
[3,3,3,3,2,1,1,1]
=> [3,3,3,2,1,1,1]
=> [4,4,2,2,2]
=> ? = 7
[3,3,3,3,1,1,1,1,1]
=> [3,3,3,1,1,1,1,1]
=> [6,3,2,2,1]
=> ? = 6
[3,3,3,2,1,1,1,1,1,1]
=> [3,3,2,1,1,1,1,1,1]
=> [6,3,3,1,1]
=> ? = 5
[3,3,3,1,1,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1,1,1]
=> [6,4,2,1,1]
=> ? = 4
[3,3,2,2,2,2,2,1]
=> [3,2,2,2,2,2,1]
=> [4,3,3,3,1]
=> ? = 7
[3,3,2,2,2,2,1,1,1]
=> [3,2,2,2,2,1,1,1]
=> [6,4,4]
=> ? = 6
[3,3,2,2,2,1,1,1,1,1]
=> [3,2,2,2,1,1,1,1,1]
=> [5,3,3,3]
=> ? = 5
[3,3,2,2,1,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [6,3,3,2]
=> ? = 4
[3,3,2,1,1,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> [6,4,2,2]
=> ? = 3
[3,3,1,1,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [6,4,3,1]
=> ? = 2
Description
The dinv defect of an integer partition.
This is the number of cells c in the diagram of an integer partition λ for which arm(c)−leg(c)∉{0,1}.
Matching statistic: St001175
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 85%●distinct values known / distinct values provided: 62%
St001175: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 85%●distinct values known / distinct values provided: 62%
Values
[1,1]
=> [2]
=> 0
[2,1]
=> [2,1]
=> 0
[1,1,1]
=> [3]
=> 0
[3,1]
=> [2,1,1]
=> 0
[2,2]
=> [2,2]
=> 1
[2,1,1]
=> [3,1]
=> 0
[1,1,1,1]
=> [4]
=> 0
[4,1]
=> [2,1,1,1]
=> 0
[3,2]
=> [2,2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> 0
[2,2,1]
=> [3,2]
=> 1
[2,1,1,1]
=> [4,1]
=> 0
[1,1,1,1,1]
=> [5]
=> 0
[5,1]
=> [2,1,1,1,1]
=> 0
[4,2]
=> [2,2,1,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> 0
[3,3]
=> [2,2,2]
=> 2
[3,2,1]
=> [3,2,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> 0
[2,2,2]
=> [3,3]
=> 2
[2,2,1,1]
=> [4,2]
=> 1
[2,1,1,1,1]
=> [5,1]
=> 0
[1,1,1,1,1,1]
=> [6]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> 0
[5,2]
=> [2,2,1,1,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> 0
[4,3]
=> [2,2,2,1]
=> 2
[4,2,1]
=> [3,2,1,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> 0
[3,3,1]
=> [3,2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> 2
[3,2,1,1]
=> [4,2,1]
=> 1
[3,1,1,1,1]
=> [5,1,1]
=> 0
[2,2,2,1]
=> [4,3]
=> 2
[2,2,1,1,1]
=> [5,2]
=> 1
[2,1,1,1,1,1]
=> [6,1]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 0
[5,3]
=> [2,2,2,1,1]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> 1
[5,1,1,1]
=> [4,1,1,1,1]
=> 0
[4,4]
=> [2,2,2,2]
=> 3
[4,3,1]
=> [3,2,2,1]
=> 2
[4,2,2]
=> [3,3,1,1]
=> 2
[4,2,1,1]
=> [4,2,1,1]
=> 1
[4,1,1,1,1]
=> [5,1,1,1]
=> 0
[3,3,2]
=> [3,3,2]
=> 3
[3,3,1,1]
=> [4,2,2]
=> 2
[16,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[15,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[15,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[14,3]
=> [2,2,2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 2
[14,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[13,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1,1]
=> ? = 3
[12,5]
=> [2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 4
[11,6]
=> [2,2,2,2,2,2,1,1,1,1,1]
=> ? = 5
[10,7]
=> [2,2,2,2,2,2,2,1,1,1]
=> ? = 6
[9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 7
[6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 10
[5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? = 10
[6,4,4,3,2,1]
=> [6,5,4,3,1,1]
=> ? = 9
[5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? = 9
[4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 9
[6,5,3,3,2,1]
=> [6,5,4,2,2,1]
=> ? = 9
[5,5,3,3,2,1]
=> [6,5,4,2,2]
=> ? = 9
[6,4,3,3,2,1]
=> [6,5,4,2,1,1]
=> ? = 8
[5,4,3,3,2,1]
=> [6,5,4,2,1]
=> ? = 8
[6,3,3,3,2,1]
=> [6,5,4,1,1,1]
=> ? = 7
[6,5,4,2,2,1]
=> [6,5,3,3,2,1]
=> ? = 9
[5,5,4,2,2,1]
=> [6,5,3,3,2]
=> ? = 9
[6,4,4,2,2,1]
=> [6,5,3,3,1,1]
=> ? = 8
[5,4,4,2,2,1]
=> [6,5,3,3,1]
=> ? = 8
[6,5,3,2,2,1]
=> [6,5,3,2,2,1]
=> ? = 8
[5,5,3,2,2,1]
=> [6,5,3,2,2]
=> ? = 8
[6,4,3,2,2,1]
=> [6,5,3,2,1,1]
=> ? = 7
[6,5,2,2,2,1]
=> [6,5,2,2,2,1]
=> ? = 7
[6,5,4,3,1,1]
=> [6,4,4,3,2,1]
=> ? = 9
[5,5,4,3,1,1]
=> [6,4,4,3,2]
=> ? = 9
[6,4,4,3,1,1]
=> [6,4,4,3,1,1]
=> ? = 8
[5,4,4,3,1,1]
=> [6,4,4,3,1]
=> ? = 8
[6,5,3,3,1,1]
=> [6,4,4,2,2,1]
=> ? = 8
[5,5,3,3,1,1]
=> [6,4,4,2,2]
=> ? = 8
[6,4,3,3,1,1]
=> [6,4,4,2,1,1]
=> ? = 7
[6,5,4,2,1,1]
=> [6,4,3,3,2,1]
=> ? = 8
[5,5,4,2,1,1]
=> [6,4,3,3,2]
=> ? = 8
[6,4,4,2,1,1]
=> [6,4,3,3,1,1]
=> ? = 7
[6,5,3,2,1,1]
=> [6,4,3,2,2,1]
=> ? = 7
[6,5,4,1,1,1]
=> [6,3,3,3,2,1]
=> ? = 7
[6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 10
[5,5,4,3,2]
=> [5,5,4,3,2]
=> ? = 10
[6,4,4,3,2]
=> [5,5,4,3,1,1]
=> ? = 9
[5,4,4,3,2]
=> [5,5,4,3,1]
=> ? = 9
[6,5,3,3,2]
=> [5,5,4,2,2,1]
=> ? = 9
[5,5,3,3,2]
=> [5,5,4,2,2]
=> ? = 9
[6,4,3,3,2]
=> [5,5,4,2,1,1]
=> ? = 8
[6,5,4,2,2]
=> [5,5,3,3,2,1]
=> ? = 9
[5,5,4,2,2]
=> [5,5,3,3,2]
=> ? = 9
[6,4,4,2,2]
=> [5,5,3,3,1,1]
=> ? = 8
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 88%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 88%
Values
[1,1]
=> [1]
=> 10 => 01 => 0
[2,1]
=> [1]
=> 10 => 01 => 0
[1,1,1]
=> [1,1]
=> 110 => 011 => 0
[3,1]
=> [1]
=> 10 => 01 => 0
[2,2]
=> [2]
=> 100 => 010 => 1
[2,1,1]
=> [1,1]
=> 110 => 011 => 0
[1,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[4,1]
=> [1]
=> 10 => 01 => 0
[3,2]
=> [2]
=> 100 => 010 => 1
[3,1,1]
=> [1,1]
=> 110 => 011 => 0
[2,2,1]
=> [2,1]
=> 1010 => 0101 => 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 0
[5,1]
=> [1]
=> 10 => 01 => 0
[4,2]
=> [2]
=> 100 => 010 => 1
[4,1,1]
=> [1,1]
=> 110 => 011 => 0
[3,3]
=> [3]
=> 1000 => 0100 => 2
[3,2,1]
=> [2,1]
=> 1010 => 0101 => 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[2,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[2,2,1,1]
=> [2,1,1]
=> 10110 => 01011 => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011111 => 0
[6,1]
=> [1]
=> 10 => 01 => 0
[5,2]
=> [2]
=> 100 => 010 => 1
[5,1,1]
=> [1,1]
=> 110 => 011 => 0
[4,3]
=> [3]
=> 1000 => 0100 => 2
[4,2,1]
=> [2,1]
=> 1010 => 0101 => 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[3,3,1]
=> [3,1]
=> 10010 => 01001 => 2
[3,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[3,2,1,1]
=> [2,1,1]
=> 10110 => 01011 => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 0
[2,2,2,1]
=> [2,2,1]
=> 11010 => 01101 => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 010111 => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011111 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111111 => 0
[7,1]
=> [1]
=> 10 => 01 => 0
[6,2]
=> [2]
=> 100 => 010 => 1
[6,1,1]
=> [1,1]
=> 110 => 011 => 0
[5,3]
=> [3]
=> 1000 => 0100 => 2
[5,2,1]
=> [2,1]
=> 1010 => 0101 => 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[4,4]
=> [4]
=> 10000 => 01000 => 3
[4,3,1]
=> [3,1]
=> 10010 => 01001 => 2
[4,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[4,2,1,1]
=> [2,1,1]
=> 10110 => 01011 => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 0
[3,3,2]
=> [3,2]
=> 10100 => 01010 => 3
[3,3,1,1]
=> [3,1,1]
=> 100110 => 010011 => 2
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? = 0
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 0111011111 => ? = 3
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01101111111 => ? = 2
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? => ? = 0
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? => ? = 4
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? => ? = 4
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> 10101111110 => 01010111111 => ? = 3
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? = 2
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 0111011111 => ? = 3
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01101111111 => ? = 2
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? = 0
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => 0111101111 => ? = 4
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? => ? = 3
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => 011011111111 => ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? => ? = 1
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111110 => ? => ? = 0
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> 1010011110 => ? => ? = 5
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> 1001101110 => ? => ? = 5
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> 10010111110 => 01001011111 => ? = 4
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? = 3
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? => ? = 4
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? => ? = 4
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> 10101111110 => 01010111111 => ? = 3
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? = 2
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 0111011111 => ? = 3
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01101111111 => ? = 2
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? = 0
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> 1101011110 => ? => ? = 5
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => 01100111111 => ? = 4
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> 1011101110 => ? => ? = 5
[3,3,2,2,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1]
=> 10110111110 => ? => ? = 4
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> 101011111110 => ? => ? = 3
[3,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> 1001111111110 => ? => ? = 2
[3,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => 0111101111 => ? = 4
[3,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? => ? = 3
[3,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => 011011111111 => ? = 2
[3,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? => ? = 1
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? => ? = 0
[2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> 1111101110 => ? => ? = 5
[2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> 11110111110 => 01111011111 => ? = 4
[2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> 111011111110 => ? => ? = 3
[2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> 1101111111110 => ? => ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> 10111111111110 => ? => ? = 1
[2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111110 => ? => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111110 => ? => ? = 0
Description
The number of inversions of a binary word.
Matching statistic: St000369
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 81%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 81%
Values
[1,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[2,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[4,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[3,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[5,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,2,2]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0
[6,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[5,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[7,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[6,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,3,2]
=> [3,2]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[4,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[5,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[4,4,4]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 6
[4,4,3,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 5
[6,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[5,4,4]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 6
[5,4,3,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 5
[4,4,4,1]
=> [4,4,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[4,4,3,2]
=> [4,3,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[4,4,3,1,1]
=> [4,3,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 5
[7,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[6,4,4]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 6
[6,4,3,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 5
[5,5,4]
=> [5,4]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 7
[5,4,4,1]
=> [4,4,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[5,4,3,2]
=> [4,3,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[5,4,3,1,1]
=> [4,3,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 5
[4,4,4,2]
=> [4,4,2]
=> [8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[4,4,4,1,1]
=> [4,4,1,1]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 6
[4,4,3,2,1]
=> [4,3,2,1]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 6
[4,4,3,1,1,1]
=> [4,3,1,1,1]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 5
[8,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[7,4,4]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 6
[7,4,3,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 5
[6,5,4]
=> [5,4]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 7
[6,4,4,1]
=> [4,4,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[6,4,3,2]
=> [4,3,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[6,4,3,1,1]
=> [4,3,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 5
[5,5,5]
=> [5,5]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[5,5,4,1]
=> [5,4,1]
=> [9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 7
[5,5,3,2]
=> [5,3,2]
=> [7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[5,4,4,2]
=> [4,4,2]
=> [8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[5,4,4,1,1]
=> [4,4,1,1]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 6
[5,4,3,2,1]
=> [4,3,2,1]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 6
[5,4,3,1,1,1]
=> [4,3,1,1,1]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 5
[4,4,4,2,1]
=> [4,4,2,1]
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 7
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 6
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 6
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 5
[3,3,3,3,3]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 7
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 6
[9,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[8,8]
=> [8]
=> [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
[8,4,4]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 6
[8,4,3,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 5
[7,5,4]
=> [5,4]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 7
[7,4,4,1]
=> [4,4,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[7,4,3,2]
=> [4,3,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[7,4,3,1,1]
=> [4,3,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 5
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
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 94%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 94%
Values
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1]
=> [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
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,1,1,1]
=> [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]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2,1,1,1]
=> [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
[2,1,1,1,1,1]
=> [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
[1,1,1,1,1,1,1]
=> [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
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 0
[3,3,3,2,1,1,1]
=> [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
[3,3,2,2,1,1,1,1]
=> [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
[3,3,2,1,1,1,1,1,1]
=> [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
[3,3,1,1,1,1,1,1,1,1]
=> [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
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [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
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 0
[4,4,4,1,1,1]
=> [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
[4,4,3,2,1,1]
=> [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
[4,4,3,1,1,1,1]
=> [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
[4,4,2,2,1,1,1]
=> [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
[4,4,2,1,1,1,1,1]
=> [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
[4,4,1,1,1,1,1,1,1]
=> [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
[4,3,3,2,1,1,1]
=> [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
[4,3,2,2,1,1,1,1]
=> [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
[4,3,2,1,1,1,1,1,1]
=> [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
[4,3,1,1,1,1,1,1,1,1]
=> [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
[3,3,3,2,2,1,1]
=> [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
[3,3,3,2,1,1,1,1]
=> [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
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[3,3,2,2,2,1,1,1]
=> [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
[3,3,2,2,1,1,1,1,1]
=> [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
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 3
[3,3,1,1,1,1,1,1,1,1,1]
=> [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
[3,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[3,2,1,1,1,1,1,1,1,1,1,1]
=> [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
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 0
[2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 1
[2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 0
[5,5,3,2,1]
=> [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
[5,5,3,1,1,1]
=> [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
[5,5,2,2,1,1]
=> [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
[5,5,2,1,1,1,1]
=> [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
[5,5,1,1,1,1,1,1]
=> [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
[5,4,4,1,1,1]
=> [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
[5,4,3,2,1,1]
=> [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
[5,4,3,1,1,1,1]
=> [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
[5,4,2,2,1,1,1]
=> [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
[5,4,2,1,1,1,1,1]
=> [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
[5,4,1,1,1,1,1,1,1]
=> [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
[5,3,3,2,1,1,1]
=> [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
[5,3,2,2,1,1,1,1]
=> [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
[5,3,2,1,1,1,1,1,1]
=> [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
[5,3,1,1,1,1,1,1,1,1]
=> [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
[4,4,4,2,1,1]
=> [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
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000738
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 69% ●values known / values provided: 72%●distinct values known / distinct values provided: 69%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 69% ●values known / values provided: 72%●distinct values known / distinct values provided: 69%
Values
[1,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[2,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[3,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[2,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[4,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[3,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[5,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[4,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[4,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[3,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
[6,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[5,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[5,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[4,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 1 = 0 + 1
[7,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[6,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[6,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[5,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[5,2,1]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[4,4]
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 5 + 1
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? = 4 + 1
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? = 3 + 1
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? = 2 + 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [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
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 5 + 1
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? = 4 + 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 6 + 1
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 5 + 1
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? = 4 + 1
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? = 3 + 1
[3,3,1,1,1,1,1,1,1,1]
=> [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
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 5 + 1
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? = 4 + 1
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? = 3 + 1
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? = 2 + 1
[3,2,1,1,1,1,1,1,1,1,1]
=> [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
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? = 5 + 1
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> [[1,2,3,4,9,10,11,12],[5,6,7,8]]
=> ? = 4 + 1
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 3 + 1
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 2 + 1
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [11,1]
=> [[1,3,4,5,6,7,8,9,10,11,12],[2]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? = 0 + 1
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? = 5 + 1
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? = 5 + 1
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ? = 4 + 1
[4,4,1,1,1,1,1,1,1]
=> [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
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 5 + 1
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? = 4 + 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 6 + 1
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 5 + 1
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? = 4 + 1
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? = 3 + 1
[4,3,1,1,1,1,1,1,1,1]
=> [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
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 5 + 1
[4,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? = 4 + 1
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? = 3 + 1
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? = 2 + 1
[4,2,1,1,1,1,1,1,1,1,1]
=> [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
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[3,3,3,3,3]
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 8 + 1
[3,3,3,3,1,1,1]
=> [3,3,3,1,1,1]
=> [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? = 6 + 1
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 7 + 1
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? = 5 + 1
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 4 + 1
[3,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 6 + 1
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [7,4,1]
=> [[1,3,4,5,10,11,12],[2,7,8,9],[6]]
=> ? = 5 + 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: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 69% ●values known / values provided: 72%●distinct values known / distinct values provided: 69%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 69% ●values known / values provided: 72%●distinct values known / distinct values provided: 69%
Values
[1,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[2,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[3,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[2,2]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[4,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3,2]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[5,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[4,2]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[3,3]
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[6,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[5,2]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[4,3]
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1 = 0 + 1
[7,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[6,2]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[5,3]
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 0 + 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 5 + 1
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 4 + 1
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 3 + 1
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2 + 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 0 + 1
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? = 5 + 1
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? = 4 + 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 6 + 1
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 5 + 1
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? = 4 + 1
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 3 + 1
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2 + 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 5 + 1
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 4 + 1
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 3 + 1
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2 + 1
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 0 + 1
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 5 + 1
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4 + 1
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 3 + 1
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2 + 1
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 1 + 1
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 0 + 1
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? = 5 + 1
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 5 + 1
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 4 + 1
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 3 + 1
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? = 5 + 1
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? = 4 + 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 6 + 1
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 5 + 1
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? = 4 + 1
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 3 + 1
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2 + 1
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 5 + 1
[4,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 4 + 1
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 3 + 1
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2 + 1
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 0 + 1
[3,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 8 + 1
[3,3,3,3,1,1,1]
=> [3,3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11],[12]]
=> ? = 6 + 1
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12]]
=> ? = 7 + 1
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 5 + 1
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 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: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 69%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 69%
Values
[1,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[2,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[3,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[2,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[4,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[3,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[5,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[4,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[3,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[6,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[5,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 0
[7,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[6,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[5,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 2
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[2,2,2,2,2,2,1]
=> [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
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ?
=> ? = 4
[2,2,2,2,1,1,1,1,1]
=> [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
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ?
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1]
=> [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
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[3,3,3,3,2]
=> [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
[3,3,3,3,1,1]
=> [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
[3,3,3,2,2,1]
=> [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
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 5
[3,3,3,1,1,1,1,1]
=> [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
[3,3,2,2,2,2]
=> [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
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 5
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ?
=> ? = 4
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ?
=> ? = 3
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? = 2
[3,2,2,2,2,2,1]
=> [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
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ?
=> ? = 4
[3,2,2,2,1,1,1,1,1]
=> [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
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ?
=> ? = 2
[3,2,1,1,1,1,1,1,1,1,1]
=> [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
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ?
=> ? = 5
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ?
=> ? = 4
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ?
=> ? = 3
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? = 0
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 8
[4,4,4,2,1]
=> [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
[4,4,4,1,1,1]
=> [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
[4,4,3,3,1]
=> [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
[4,4,3,2,2]
=> [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
[4,4,3,2,1,1]
=> [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
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ?
=> ? = 5
[4,4,2,2,2,1]
=> [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
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ?
=> ? = 5
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ?
=> ? = 4
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ?
=> ? = 3
[4,3,3,3,2]
=> [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
[4,3,3,3,1,1]
=> [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
[4,3,3,2,2,1]
=> [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
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 5
[4,3,3,1,1,1,1,1]
=> [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
[4,3,2,2,2,2]
=> [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
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 5
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ?
=> ? = 4
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ?
=> ? = 3
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? = 2
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.
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000074The number of special entries. St000211The rank of the set partition. St000502The number of successions of a set partitions. St001298The number of repeated entries in the Lehmer code of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000065The number of entries equal to -1 in an alternating sign matrix. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St000223The number of nestings in the permutation. St000833The comajor index of a permutation. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St000463The number of admissible inversions of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St000039The number of crossings of a permutation. St000355The number of occurrences of the pattern 21-3. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000233The number of nestings of a set partition. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St001394The genus of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000204The number of internal nodes of a binary tree. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000456The monochromatic index of a connected graph. St000537The cutwidth of a graph. St001271The competition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001883The mutual visibility number of a graph. St001323The independence gap of a graph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St001644The dimension of a graph. St001330The hat guessing number of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000624The normalized sum of the minimal distances to a greater element. St000779The tier of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St000450The number of edges minus the number of vertices plus 2 of a graph. St000991The number of right-to-left minima of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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