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Your data matches 72 different statistics following compositions of up to 3 maps.
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Matching statistic: St001175
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 0
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 1
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 0
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> 1
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 1
[4,1,1]
=> 0
[3,3]
=> 2
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 2
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 0
[5,2]
=> 1
[5,1,1]
=> 0
[4,3]
=> 2
[4,2,1]
=> 1
[4,1,1,1]
=> 0
[3,3,1]
=> 2
[3,2,2]
=> 2
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 0
[6,2]
=> 1
[6,1,1]
=> 0
[5,3]
=> 2
[5,2,1]
=> 1
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: St001176
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> []
=> []
=> ? = 0
[2]
=> [1,1]
=> [1]
=> [1]
=> 0
[1,1]
=> [2]
=> []
=> []
=> ? = 0
[3]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[2,1]
=> [2,1]
=> [1]
=> [1]
=> 0
[1,1,1]
=> [3]
=> []
=> []
=> ? = 0
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[2,1,1]
=> [3,1]
=> [1]
=> [1]
=> 0
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 0
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,2]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,1]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [1]
=> 0
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 0
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,3]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1]
=> 2
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1]
=> 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1]
=> 0
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 0
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1]
=> 2
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1]
=> 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [3,2]
=> 2
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 2
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [2,2,1]
=> 3
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [14]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [15]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [16]
=> []
=> []
=> ? = 0
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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> ?
=> ? = 0
[2]
=> []
=> []
=> ?
=> ? = 0
[1,1]
=> [1]
=> [1]
=> []
=> 0
[3]
=> []
=> []
=> ?
=> ? = 0
[2,1]
=> [1]
=> [1]
=> []
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[4]
=> []
=> []
=> ?
=> ? = 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
[5]
=> []
=> []
=> ?
=> ? = 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
[6]
=> []
=> []
=> ?
=> ? = 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
[7]
=> []
=> []
=> ?
=> ? = 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
[8]
=> []
=> []
=> ?
=> ? = 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
[9]
=> []
=> []
=> ?
=> ? = 0
[10]
=> []
=> []
=> ?
=> ? = 0
[11]
=> []
=> []
=> ?
=> ? = 0
[12]
=> []
=> []
=> ?
=> ? = 0
[13]
=> []
=> []
=> ?
=> ? = 0
[14]
=> []
=> []
=> ?
=> ? = 0
[15]
=> []
=> []
=> ?
=> ? = 0
[16]
=> []
=> []
=> ?
=> ? = 0
[17]
=> []
=> []
=> ?
=> ? = 0
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: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> 0
[2]
=> []
=> []
=> 0
[1,1]
=> [1]
=> [1]
=> 0
[3]
=> []
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> 0
[4]
=> []
=> []
=> 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
[5]
=> []
=> []
=> 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
[6]
=> []
=> []
=> 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
[7]
=> []
=> []
=> 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
[8]
=> []
=> []
=> 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
[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 $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
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: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[4]
=> []
=> []
=> []
=> ? = 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
[5]
=> []
=> []
=> []
=> ? = 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
[6]
=> []
=> []
=> []
=> ? = 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
[7]
=> []
=> []
=> []
=> ? = 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
[8]
=> []
=> []
=> []
=> ? = 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
[9]
=> []
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> []
=> ? = 0
[11]
=> []
=> []
=> []
=> ? = 0
[4,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[12]
=> []
=> []
=> []
=> ? = 0
[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
[13]
=> []
=> []
=> []
=> ? = 0
[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
[14]
=> []
=> []
=> []
=> ? = 0
[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
[15]
=> []
=> []
=> []
=> ? = 0
[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
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: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[4]
=> []
=> []
=> []
=> ? = 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
[5]
=> []
=> []
=> []
=> ? = 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
[6]
=> []
=> []
=> []
=> ? = 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
[7]
=> []
=> []
=> []
=> ? = 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
[8]
=> []
=> []
=> []
=> ? = 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
[9]
=> []
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> []
=> ? = 0
[11]
=> []
=> []
=> []
=> ? = 0
[12]
=> []
=> []
=> []
=> ? = 0
[13]
=> []
=> []
=> []
=> ? = 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 0
[14]
=> []
=> []
=> []
=> ? = 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
[15]
=> []
=> []
=> []
=> ? = 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
[16]
=> []
=> []
=> []
=> ? = 0
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
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: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> => ? => ? = 0
[2]
=> []
=> => ? => ? = 0
[1,1]
=> [1]
=> 10 => 01 => 0
[3]
=> []
=> => ? => ? = 0
[2,1]
=> [1]
=> 10 => 01 => 0
[1,1,1]
=> [1,1]
=> 110 => 011 => 0
[4]
=> []
=> => ? => ? = 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
[5]
=> []
=> => ? => ? = 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
[6]
=> []
=> => ? => ? = 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
[7]
=> []
=> => ? => ? = 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
[8]
=> []
=> => ? => ? = 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
[9]
=> []
=> => ? => ? = 0
[10]
=> []
=> => ? => ? = 0
[11]
=> []
=> => ? => ? = 0
[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]
=> 111111111110 => ? => ? = 0
[13]
=> []
=> => ? => ? = 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
[14]
=> []
=> => ? => ? = 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
[15]
=> []
=> => ? => ? = 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
Description
The number of inversions of a binary word.
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: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> ? = 0 + 1
[2]
=> []
=> []
=> []
=> ? = 0 + 1
[1,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3]
=> []
=> []
=> []
=> ? = 0 + 1
[2,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[4]
=> []
=> []
=> []
=> ? = 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
[5]
=> []
=> []
=> []
=> ? = 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
[6]
=> []
=> []
=> []
=> ? = 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
[7]
=> []
=> []
=> []
=> ? = 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
[8]
=> []
=> []
=> []
=> ? = 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
[9]
=> []
=> []
=> []
=> ? = 0 + 1
[10]
=> []
=> []
=> []
=> ? = 0 + 1
[11]
=> []
=> []
=> []
=> ? = 0 + 1
[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]
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 0 + 1
[13]
=> []
=> []
=> []
=> ? = 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
[14]
=> []
=> []
=> []
=> ? = 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
[15]
=> []
=> []
=> []
=> ? = 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
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)
(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: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 90%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 90%
Values
[1]
=> []
=> []
=> []
=> 0
[2]
=> []
=> []
=> []
=> 0
[1,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[3]
=> []
=> []
=> []
=> 0
[2,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[4]
=> []
=> []
=> []
=> 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
[5]
=> []
=> []
=> []
=> 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
[6]
=> []
=> []
=> []
=> 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
[7]
=> []
=> []
=> []
=> 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
[8]
=> []
=> []
=> []
=> 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
[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 $i$.
Matching statistic: St000245
(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
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 90%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 90%
Values
[1]
=> []
=> []
=> [] => 0
[2]
=> []
=> []
=> [] => 0
[1,1]
=> [1]
=> [[1]]
=> [1] => 0
[3]
=> []
=> []
=> [] => 0
[2,1]
=> [1]
=> [[1]]
=> [1] => 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4]
=> []
=> []
=> [] => 0
[3,1]
=> [1]
=> [[1]]
=> [1] => 0
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[5]
=> []
=> []
=> [] => 0
[4,1]
=> [1]
=> [[1]]
=> [1] => 0
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[6]
=> []
=> []
=> [] => 0
[5,1]
=> [1]
=> [[1]]
=> [1] => 0
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[7]
=> []
=> []
=> [] => 0
[6,1]
=> [1]
=> [[1]]
=> [1] => 0
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
[8]
=> []
=> []
=> [] => 0
[7,1]
=> [1]
=> [[1]]
=> [1] => 0
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 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]]
=> ? => ? = 0
[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
[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
[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
[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
[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
[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
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 7
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? = 6
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 6
[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
[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
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? = 6
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[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
[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
[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
[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
[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
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 8
[4,4,4,2,1]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [11,9,10,5,6,7,8,1,2,3,4] => ? = 7
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [11,10,9,5,6,7,8,1,2,3,4] => ? = 6
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 7
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [10,11,8,9,5,6,7,1,2,3,4] => ? = 7
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 6
[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
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,5,6,1,2,3,4] => ? = 6
[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
[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
[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
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 7
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? = 6
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 6
[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
[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
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? = 6
[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
[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
[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
Description
The number of ascents of a permutation.
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001727The number of invisible inversions of a permutation. St000463The number of admissible inversions of a permutation. St000355The number of occurrences of the pattern 21-3. St000039The number of crossings of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. 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 $3$. St000233The number of nestings of a set partition. St001394The genus of a permutation. St000565The major index of a set partition. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000562The number of internal points 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. 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. 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. St001323The independence gap of a graph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000358The number of occurrences of the pattern 31-2. 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. St001811The Castelnuovo-Mumford regularity of a permutation. St000624The normalized sum of the minimal distances to a greater element. 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. St000317The cycle descent number of a permutation. St000779The tier of a permutation. 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. St001866The nesting alignments of a signed permutation.
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