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Your data matches 177 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)
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 1
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 2
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 3
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 2
[3,1,1]
=> 2
[2,2,1]
=> 3
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 4
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 2
[3,3]
=> 3
[3,2,1]
=> 3
[3,1,1,1]
=> 3
[2,2,2]
=> 4
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 5
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 2
[4,3]
=> 3
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 4
[3,2,2]
=> 4
[3,2,1,1]
=> 4
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 5
[2,2,1,1,1]
=> 5
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 6
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 2
[5,3]
=> 3
[5,2,1]
=> 3
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: St001034
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> 0
[2]
=> []
=> []
=> []
=> 0
[1,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[3]
=> []
=> []
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[4]
=> []
=> []
=> []
=> 0
[3,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[5]
=> []
=> []
=> []
=> 0
[4,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[2,2,1]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[6]
=> []
=> []
=> []
=> 0
[5,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[3,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[3,2,1]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
[7]
=> []
=> []
=> []
=> 0
[6,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[4,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[4,2,1]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[3,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[8]
=> []
=> []
=> []
=> 0
[7,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[5,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[5,2,1]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11}
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11}
[4,4,4,2,1]
=> [4,4,2,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[4,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[3,3,3,3,3]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,12,12,12,13,13}
[5,5,4,1,1]
=> [5,4,1,1]
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,5,3,3]
=> [5,3,3]
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,5,3,2,1]
=> [5,3,2,1]
=> [7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,5,2,2,1,1]
=> [5,2,2,1,1]
=> [6,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,5,2,1,1,1,1]
=> [5,2,1,1,1,1]
=> [6,3,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,4,4,2,1]
=> [4,4,2,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,4,4,1,1,1]
=> [4,4,1,1,1]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,4,3,2,1,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,3,3,2,2,1]
=> [3,3,2,2,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,4,2,2]
=> [4,4,2,2]
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,4,2,1,1]
=> [4,4,2,1,1]
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,4,1,1,1,1]
=> [4,4,1,1,1,1]
=> [8,3,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,3,2,1,1,1]
=> [4,3,2,1,1,1]
=> [6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,3,1,1,1,1,1]
=> [4,3,1,1,1,1,1]
=> [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,2,2,2,1,1]
=> [4,2,2,2,1,1]
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,3,3,3,3]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,3,3,3,2,1]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[3,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,3,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[3,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> [7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[3,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[3,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [6,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[2,2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,2,1,1]
=> [6,5,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000293
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 94%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 94%
Values
[1]
=> []
=> []
=> => ? = 0
[2]
=> []
=> []
=> => ? = 0
[1,1]
=> [1]
=> [1]
=> 10 => 1
[3]
=> []
=> []
=> => ? = 0
[2,1]
=> [1]
=> [1]
=> 10 => 1
[1,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[4]
=> []
=> []
=> => ? = 0
[3,1]
=> [1]
=> [1]
=> 10 => 1
[2,2]
=> [2]
=> [1,1]
=> 110 => 2
[2,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[5]
=> []
=> []
=> => ? = 0
[4,1]
=> [1]
=> [1]
=> 10 => 1
[3,2]
=> [2]
=> [1,1]
=> 110 => 2
[3,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[2,2,1]
=> [2,1]
=> [3]
=> 1000 => 3
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 10010 => 4
[6]
=> []
=> []
=> => ? = 0
[5,1]
=> [1]
=> [1]
=> 10 => 1
[4,2]
=> [2]
=> [1,1]
=> 110 => 2
[4,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[3,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[3,2,1]
=> [2,1]
=> [3]
=> 1000 => 3
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[2,2,2]
=> [2,2]
=> [4]
=> 10000 => 4
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 10010 => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 10100 => 5
[7]
=> []
=> []
=> => ? = 0
[6,1]
=> [1]
=> [1]
=> 10 => 1
[5,2]
=> [2]
=> [1,1]
=> 110 => 2
[5,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[4,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[4,2,1]
=> [2,1]
=> [3]
=> 1000 => 3
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 10110 => 4
[3,2,2]
=> [2,2]
=> [4]
=> 10000 => 4
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 10010 => 4
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 11010 => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 100110 => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 10100 => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 101010 => 6
[8]
=> []
=> []
=> => ? = 0
[7,1]
=> [1]
=> [1]
=> 10 => 1
[6,2]
=> [2]
=> [1,1]
=> 110 => 2
[6,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[5,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[5,2,1]
=> [2,1]
=> [3]
=> 1000 => 3
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[4,4]
=> [4]
=> [1,1,1,1]
=> 11110 => 4
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 10110 => 4
[4,2,2]
=> [2,2]
=> [4]
=> 10000 => 4
[4,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 4
[4,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 10010 => 4
[3,3,2]
=> [3,2]
=> [5]
=> 100000 => 5
[3,3,1,1]
=> [3,1,1]
=> [4,1]
=> 100010 => 5
[9]
=> []
=> []
=> => ? = 0
[10]
=> []
=> []
=> => ? = 0
[11]
=> []
=> []
=> => ? = 0
[12]
=> []
=> []
=> => ? = 0
[13]
=> []
=> []
=> => ? = 0
[14]
=> []
=> []
=> => ? ∊ {0,11}
[3,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> 100000111110 => ? ∊ {0,11}
[15]
=> []
=> []
=> => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[4,4,4,2,1]
=> [4,4,2,1]
=> [8,3]
=> 1000001000 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> 1000001100 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> 1000010010 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[4,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> 100000111110 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[3,3,3,3,3]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> 1000000111110 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> 100001011110 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> 10000100110 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [6,4,1,1]
=> 1001000110 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> 10000111010 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> 10010001110 => ? ∊ {0,11,11,11,11,12,12,12,12,13,13}
[16]
=> []
=> []
=> => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,5,5,1]
=> [5,5,1]
=> [10,1]
=> 100000000010 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,5,4,2]
=> [5,4,2]
=> [9,1,1]
=> 100000000110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,5,3,3]
=> [5,3,3]
=> [8,1,1,1]
=> 100000001110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,5,2,2,1,1]
=> [5,2,2,1,1]
=> [6,2,2,1]
=> 1000011010 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,5,2,1,1,1,1]
=> [5,2,1,1,1,1]
=> [6,3,1,1]
=> 1000100110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,4,4,2,1]
=> [4,4,2,1]
=> [8,3]
=> 1000001000 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,4,3,2,1,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> 1000001100 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> 1000010010 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[5,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> 100000111110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,4,2,2]
=> [4,4,2,2]
=> [8,4]
=> 1000010000 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,4,2,1,1]
=> [4,4,2,1,1]
=> [8,2,2]
=> 10000001100 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,4,1,1,1,1]
=> [4,4,1,1,1,1]
=> [8,3,1]
=> 10000010010 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [6,2,2,1,1]
=> 10000110110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,3,2,1,1,1]
=> [4,3,2,1,1,1]
=> [6,3,1,1,1]
=> 10001001110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,3,1,1,1,1,1]
=> [4,3,1,1,1,1,1]
=> [7,3,2]
=> 1000010100 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,4,2,2,2,1,1]
=> [4,2,2,2,1,1]
=> [6,2,2,2]
=> 1000011100 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,3,3,3,3]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> 1000000111110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,3,3,3,2,1]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> 100001011110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> 10000100110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[4,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [6,4,1,1]
=> 1001000110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> 1000001011110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[3,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,3,1,1,1,1]
=> 100010011110 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
[3,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> [7,5,1]
=> 1001000010 => ? ∊ {0,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,14,14}
Description
The number of inversions of a binary word.
Matching statistic: St000246
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 84%●distinct values known / distinct values provided: 81%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 84%●distinct values known / distinct values provided: 81%
Values
[1]
=> []
=> []
=> [] => 0
[2]
=> []
=> []
=> [] => 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[3]
=> []
=> []
=> [] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[4]
=> []
=> []
=> [] => 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[5]
=> []
=> []
=> [] => 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[6]
=> []
=> []
=> [] => 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 5
[7]
=> []
=> []
=> [] => 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 6
[8]
=> []
=> []
=> [] => 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ? ∊ {11,11,11,11,11,11,12}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,8,3,7,6,5,4,2] => ? ∊ {11,11,11,11,11,11,12}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,9,8,4,7,6,5,3,2] => ? ∊ {11,11,11,11,11,11,12}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,5,7,6,4,3,2] => ? ∊ {11,11,11,11,11,11,12}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,6,7,5,4,3,2] => ? ∊ {11,11,11,11,11,11,12}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,12}
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,12}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,8,4,7,5,6,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,9,8,5,6,7,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,10,9,8,6,5,7,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,8,3,7,6,5,4,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,9,8,4,7,6,5,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,5,7,6,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,6,7,5,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,8,7,6,5,4,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,9,4,8,7,6,5,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,10,9,5,8,7,6,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,6,8,7,5,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,12,11,10,9,7,8,6,5,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,7,4,3,5,6,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,7,3,5,4,6,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,8,6,4,5,7,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,9,8,6,5,4,7,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,8,4,7,5,6,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,9,8,5,6,7,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,10,9,8,6,5,7,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,8,3,7,6,5,4,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,9,8,4,7,6,5,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,5,7,6,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,6,7,5,4,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,3,1,1,1]
=> [3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,3,7,6,5,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,7,4,6,5,3] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,8,4,5,7,6,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,9,8,5,4,7,6,3,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
Description
The number of non-inversions of a permutation.
For a permutation of {1,…,n}, this is given by \operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi).
Matching statistic: St000018
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 84%●distinct values known / distinct values provided: 81%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 84%●distinct values known / distinct values provided: 81%
Values
[1]
=> []
=> []
=> [] => 0
[2]
=> []
=> []
=> [] => 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> []
=> []
=> [] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4]
=> []
=> []
=> [] => 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[5]
=> []
=> []
=> [] => 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[6]
=> []
=> []
=> [] => 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[7]
=> []
=> []
=> [] => 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 6
[8]
=> []
=> []
=> [] => 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? ∊ {11,11,11,11,11,11,12}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? ∊ {11,11,11,11,11,11,12}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? ∊ {11,11,11,11,11,11,12}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? ∊ {11,11,11,11,11,11,12}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? ∊ {11,11,11,11,11,11,12}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,12}
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,12}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,7,2,8,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,9,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,10,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,11,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,12,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,7,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,7,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [5,2,3,4,6,7,8,9,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,3,1,1,1]
=> [3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,1,2] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,6,7,8,1] => ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions (i,i+1) needed to write \pi. Thus, it is also the Coxeter length of \pi.
Matching statistic: St000290
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 88%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 88%
Values
[1]
=> []
=> => ? => ? = 0
[2]
=> []
=> => ? => ? = 0
[1,1]
=> [1]
=> 10 => 10 => 1
[3]
=> []
=> => ? => ? = 0
[2,1]
=> [1]
=> 10 => 10 => 1
[1,1,1]
=> [1,1]
=> 110 => 110 => 2
[4]
=> []
=> => ? => ? = 0
[3,1]
=> [1]
=> 10 => 10 => 1
[2,2]
=> [2]
=> 100 => 010 => 2
[2,1,1]
=> [1,1]
=> 110 => 110 => 2
[1,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[5]
=> []
=> => ? => ? = 0
[4,1]
=> [1]
=> 10 => 10 => 1
[3,2]
=> [2]
=> 100 => 010 => 2
[3,1,1]
=> [1,1]
=> 110 => 110 => 2
[2,2,1]
=> [2,1]
=> 1010 => 0110 => 3
[2,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 4
[6]
=> []
=> => ? => ? = 0
[5,1]
=> [1]
=> 10 => 10 => 1
[4,2]
=> [2]
=> 100 => 010 => 2
[4,1,1]
=> [1,1]
=> 110 => 110 => 2
[3,3]
=> [3]
=> 1000 => 0010 => 3
[3,2,1]
=> [2,1]
=> 1010 => 0110 => 3
[3,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[2,2,2]
=> [2,2]
=> 1100 => 1010 => 4
[2,2,1,1]
=> [2,1,1]
=> 10110 => 01110 => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 5
[7]
=> []
=> => ? => ? = 0
[6,1]
=> [1]
=> 10 => 10 => 1
[5,2]
=> [2]
=> 100 => 010 => 2
[5,1,1]
=> [1,1]
=> 110 => 110 => 2
[4,3]
=> [3]
=> 1000 => 0010 => 3
[4,2,1]
=> [2,1]
=> 1010 => 0110 => 3
[4,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[3,3,1]
=> [3,1]
=> 10010 => 00110 => 4
[3,2,2]
=> [2,2]
=> 1100 => 1010 => 4
[3,2,1,1]
=> [2,1,1]
=> 10110 => 01110 => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 4
[2,2,2,1]
=> [2,2,1]
=> 11010 => 10110 => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 011110 => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => 6
[8]
=> []
=> => ? => ? = 0
[7,1]
=> [1]
=> 10 => 10 => 1
[6,2]
=> [2]
=> 100 => 010 => 2
[6,1,1]
=> [1,1]
=> 110 => 110 => 2
[5,3]
=> [3]
=> 1000 => 0010 => 3
[5,2,1]
=> [2,1]
=> 1010 => 0110 => 3
[5,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[4,4]
=> [4]
=> 10000 => 00010 => 4
[4,3,1]
=> [3,1]
=> 10010 => 00110 => 4
[4,2,2]
=> [2,2]
=> 1100 => 1010 => 4
[4,2,1,1]
=> [2,1,1]
=> 10110 => 01110 => 4
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 4
[3,3,2]
=> [3,2]
=> 10100 => 10010 => 5
[3,3,1,1]
=> [3,1,1]
=> 100110 => 001110 => 5
[9]
=> []
=> => ? => ? = 0
[10]
=> []
=> => ? => ? = 0
[11]
=> []
=> => ? => ? ∊ {0,9}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? ∊ {0,9}
[12]
=> []
=> => ? => ? ∊ {0,9,9,10,11}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0011111110 => ? ∊ {0,9,9,10,11}
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? ∊ {0,9,9,10,11}
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01111111110 => ? ∊ {0,9,9,10,11}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? ∊ {0,9,9,10,11}
[13]
=> []
=> => ? => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0011111110 => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> 10011111110 => 00111111110 => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[3,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01111111110 => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => ? => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => ? => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[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]
=> 1111111111110 => ? => ? ∊ {0,9,9,10,10,11,11,11,11,12}
[14]
=> []
=> => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[5,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0011111110 => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[5,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[4,4,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1]
=> 10001111110 => 00011111110 => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[4,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> 10011111110 => 00111111110 => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[4,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01111111110 => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> 10101111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[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 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[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 => ? => ? ∊ {0,9,9,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12,13}
[15]
=> []
=> => ? => ? ∊ {0,9,9,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,14}
[6,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0011111110 => ? ∊ {0,9,9,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,14}
[6,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? ∊ {0,9,9,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,14}
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length n with a zeros, the generating function for the major index is the q-binomial coefficient \binom{n}{a}_q.
Matching statistic: St000228
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 80%●distinct values known / distinct values provided: 69%
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 80%●distinct values known / distinct values provided: 69%
Values
[1]
=> []
=> 0
[2]
=> []
=> 0
[1,1]
=> [1]
=> 1
[3]
=> []
=> 0
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 2
[4]
=> []
=> 0
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 2
[1,1,1,1]
=> [1,1,1]
=> 3
[5]
=> []
=> 0
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 2
[2,2,1]
=> [2,1]
=> 3
[2,1,1,1]
=> [1,1,1]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> 4
[6]
=> []
=> 0
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 2
[3,3]
=> [3]
=> 3
[3,2,1]
=> [2,1]
=> 3
[3,1,1,1]
=> [1,1,1]
=> 3
[2,2,2]
=> [2,2]
=> 4
[2,2,1,1]
=> [2,1,1]
=> 4
[2,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[7]
=> []
=> 0
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 2
[4,3]
=> [3]
=> 3
[4,2,1]
=> [2,1]
=> 3
[4,1,1,1]
=> [1,1,1]
=> 3
[3,3,1]
=> [3,1]
=> 4
[3,2,2]
=> [2,2]
=> 4
[3,2,1,1]
=> [2,1,1]
=> 4
[3,1,1,1,1]
=> [1,1,1,1]
=> 4
[2,2,2,1]
=> [2,2,1]
=> 5
[2,2,1,1,1]
=> [2,1,1,1]
=> 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 6
[8]
=> []
=> 0
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 2
[5,3]
=> [3]
=> 3
[5,2,1]
=> [2,1]
=> 3
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> ? ∊ {11,11,11,11,11,11,12}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> ? ∊ {11,11,11,11,11,11,12}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,12}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,12}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,12}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,12}
[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]
=> ? ∊ {11,11,11,11,11,11,12}
[3,3,3,3,2]
=> [3,3,3,2]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[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]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[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]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13}
[4,4,4,3]
=> [4,4,3]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,4,2,1]
=> [4,4,2,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,3,3,1]
=> [4,3,3,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,3,2,2]
=> [4,3,2,2]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,3,3,2]
=> [3,3,3,2]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> ? ∊ {11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,14}
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001759
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 81%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 81%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> []
=> []
=> [] => ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4]
=> []
=> []
=> [] => ? = 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[5]
=> []
=> []
=> [] => ? = 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[6]
=> []
=> []
=> [] => ? = 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[7]
=> []
=> []
=> [] => ? = 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 6
[8]
=> []
=> []
=> [] => ? ∊ {0,7}
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,7}
[9]
=> []
=> []
=> [] => ? ∊ {0,7,8}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,7,8}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,7,8}
[10]
=> []
=> []
=> [] => ? ∊ {0,7,8,8,9}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,7,8,8,9}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,7,8,8,9}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,7,8,8,9}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,7,8,8,9}
[11]
=> []
=> []
=> [] => ? ∊ {0,7,8,8,9,9,9,10}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,7,8,8,9,9,9,10}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,7,8,8,9,9,9,10}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,7,8,8,9,9,9,10}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? ∊ {0,7,8,8,9,9,9,10}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? ∊ {0,7,8,8,9,9,9,10}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,7,8,8,9,9,9,10}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => ? ∊ {0,7,8,8,9,9,9,10}
[12]
=> []
=> []
=> [] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {0,6,7,8,8,9,9,9,9,10,10,10,10,11}
[13]
=> []
=> []
=> [] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[6,6,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[6,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[5,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => ? ∊ {0,6,7,7,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12}
Description
The Rajchgot index of a permutation.
The '''Rajchgot index''' of a permutation \sigma is the degree of the ''Grothendieck polynomial'' of \sigma. This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index [[St000004]] of the permutations smaller than or equal to \sigma in the right ''weak Bruhat order''.
Matching statistic: St000395
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 81%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 81%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[5]
=> []
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[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]
=> 4
[6]
=> []
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[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]
=> 4
[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]
=> 5
[7]
=> []
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[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]
=> 4
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5
[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]
=> 5
[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]
=> 5
[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]
=> 6
[8]
=> []
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[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]
=> 4
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[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]
=> 5
[9]
=> []
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> []
=> ? ∊ {0,9}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,9}
[11]
=> []
=> []
=> []
=> ? ∊ {0,8,9,9,10}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,9,9,10}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,9,9,10}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,9,9,10}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,9,9,10}
[12]
=> []
=> []
=> []
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,9,9,9,9,10,10,10,10,11}
[13]
=> []
=> []
=> []
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[5,5,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[5,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[5,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[3,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12}
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000376
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 81%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 81%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 5
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {6,7}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> 4
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> 5
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 4
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {6,7}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {6,7,7,8}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {6,7,7,8}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {6,7,7,8}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {6,7,7,8}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,7,7,7,8,8,8,9}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,10}
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,11}
Description
The bounce deficit of a Dyck path.
For a Dyck path D of semilength n, this is defined as
\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).
The zeta map [[Mp00032]] sends this statistic to the dinv deficit [[St000369]], both are thus equidistributed.
The following 167 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000369The dinv deficit of a Dyck path. St000719The number of alignments in a perfect matching. St000377The dinv defect of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000189The number of elements in the poset. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000507The number of ascents of a standard tableau. St000738The first entry in the last row of a standard tableau. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000728The dimension of a set partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000010The length of the partition. St000734The last entry in the first row of a standard tableau. St000157The number of descents of a standard tableau. St000141The maximum drop size of a permutation. 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. St000054The first entry of the permutation. St000074The number of special entries. St000662The staircase size of the code of a permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000288The number of ones in a binary word. St000019The cardinality of the support of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000013The height of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000833The comajor index of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000794The mak of a permutation. St001726The number of visible inversions of a permutation. St000214The number of adjacencies of a permutation. St000081The number of edges of a graph. St000539The number of odd inversions of a permutation. St000795The mad of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000439The position of the first down step of a Dyck path. St001397Number of pairs of incomparable elements in a finite poset. St000839The largest opener of a set partition. St001428The number of B-inversions of a signed permutation. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by \tau \Omega^1 composed with its inverse in the corresponding Nakayama algebra. St000004The major index of a permutation. St000030The sum of the descent differences of a permutations. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St001077The prefix exchange distance of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001869The maximum cut size of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000240The number of indices that are not small excedances. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001480The number of simple summands of the module J^2/J^3. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000971The smallest closer of a set partition. St000505The biggest entry in the block containing the 1. St000504The cardinality of the first block of a set partition. St001062The maximal size of a block of a set partition. St000502The number of successions of a set partitions. St001090The number of pop-stack-sorts needed to sort a permutation. St000503The maximal difference between two elements in a common block. St001809The index of the step at the first peak of maximal height in a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000925The number of topologically connected components of a set partition. St000211The rank of the set partition. St000024The number of double up and double down steps of a Dyck path. St000171The degree of the graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000025The number of initial rises of a Dyck path. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001580The acyclic chromatic number of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000730The maximal arc length of a set partition. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000209Maximum difference of elements in cycles. St000840The number of closers smaller than the largest opener in a perfect matching. St001298The number of repeated entries in the Lehmer code of a permutation. St000956The maximal displacement of a permutation. St000654The first descent of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001277The degeneracy of a graph. St000740The last entry of a permutation. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000670The reversal length of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001644The dimension of a graph. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000454The largest eigenvalue of a graph if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000653The last descent of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000308The height of the tree associated to a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001497The position of the largest weak excedence of a permutation. St000005The bounce statistic of a Dyck path. St000133The "bounce" of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001727The number of invisible inversions of a permutation. St000060The greater neighbor of the maximum. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000316The number of non-left-to-right-maxima of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001555The order of a signed permutation. St000673The number of non-fixed points of a permutation. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000327The number of cover relations in a poset. St001631The number of simple modules S with dim Ext^1(S,A)=1 in the incidence algebra A of the poset. St001760The number of prefix or suffix reversals needed to sort a permutation. St000422The energy of a graph, if it is integral. St000021The number of descents of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000051The size of the left subtree of a binary tree. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order.
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