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Your data matches 92 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: 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]
=> 1
[3]
=> []
=> 0
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 1
[4]
=> []
=> 0
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 1
[1,1,1,1]
=> [1,1,1]
=> 1
[5]
=> []
=> 0
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 1
[2,2,1]
=> [2,1]
=> 2
[2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 1
[6]
=> []
=> 0
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 1
[3,3]
=> [3]
=> 3
[3,2,1]
=> [2,1]
=> 2
[3,1,1,1]
=> [1,1,1]
=> 1
[2,2,2]
=> [2,2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[7]
=> []
=> 0
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 1
[4,3]
=> [3]
=> 3
[4,2,1]
=> [2,1]
=> 2
[4,1,1,1]
=> [1,1,1]
=> 1
[3,3,1]
=> [3,1]
=> 3
[3,2,2]
=> [2,2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> 2
[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
[8]
=> []
=> 0
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 1
[5,3]
=> [3]
=> 3
[5,2,1]
=> [2,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: 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]
=> [1]
=> 1
[3]
=> []
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> 1
[4]
=> []
=> []
=> 0
[3,1]
=> [1]
=> [1]
=> 1
[2,2]
=> [2]
=> [1,1]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[5]
=> []
=> []
=> 0
[4,1]
=> [1]
=> [1]
=> 1
[3,2]
=> [2]
=> [1,1]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[6]
=> []
=> []
=> 0
[5,1]
=> [1]
=> [1]
=> 1
[4,2]
=> [2]
=> [1,1]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> 3
[3,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[2,2,2]
=> [2,2]
=> [2,2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[7]
=> []
=> []
=> 0
[6,1]
=> [1]
=> [1]
=> 1
[5,2]
=> [2]
=> [1,1]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> 3
[4,2,1]
=> [2,1]
=> [2,1]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 3
[3,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[8]
=> []
=> []
=> 0
[7,1]
=> [1]
=> [1]
=> 1
[6,2]
=> [2]
=> [1,1]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> 3
[5,2,1]
=> [2,1]
=> [2,1]
=> 2
Description
The length of the partition.
Matching statistic: St000378
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 92%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 92%
Values
[1]
=> []
=> []
=> []
=> 0
[2]
=> []
=> []
=> []
=> 0
[1,1]
=> [1]
=> [1]
=> [1]
=> 1
[3]
=> []
=> []
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[4]
=> []
=> []
=> []
=> 0
[3,1]
=> [1]
=> [1]
=> [1]
=> 1
[2,2]
=> [2]
=> [1,1]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 1
[5]
=> []
=> []
=> []
=> 0
[4,1]
=> [1]
=> [1]
=> [1]
=> 1
[3,2]
=> [2]
=> [1,1]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [3]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 1
[6]
=> []
=> []
=> []
=> 0
[5,1]
=> [1]
=> [1]
=> [1]
=> 1
[4,2]
=> [2]
=> [1,1]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> [2,1]
=> 3
[3,2,1]
=> [2,1]
=> [2,1]
=> [3]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 1
[2,2,2]
=> [2,2]
=> [2,2]
=> [4]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> 1
[7]
=> []
=> []
=> []
=> 0
[6,1]
=> [1]
=> [1]
=> [1]
=> 1
[5,2]
=> [2]
=> [1,1]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> [2,1]
=> 3
[4,2,1]
=> [2,1]
=> [2,1]
=> [3]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [2,2]
=> 3
[3,2,2]
=> [2,2]
=> [2,2]
=> [4]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [5]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2
[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]
=> [1,1,1,1,1,1]
=> [6]
=> [1,1,1,1,1,1]
=> 1
[8]
=> []
=> []
=> []
=> 0
[7,1]
=> [1]
=> [1]
=> [1]
=> 1
[6,2]
=> [2]
=> [1,1]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> [2,1]
=> 3
[5,2,1]
=> [2,1]
=> [2,1]
=> [3]
=> 2
[2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2
[2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> ? = 2
[2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 2
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> [10,1,1,1]
=> ? = 3
[3,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 3
[3,3,3,3,1,1,1,1]
=> [3,3,3,1,1,1,1]
=> [7,3,3]
=> [8,1,1,1,1,1]
=> ? = 3
[3,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> [11,1,1]
=> ? = 3
[3,3,3,2,2,1,1,1]
=> [3,3,2,2,1,1,1]
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3
[3,3,3,1,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1,1]
=> [9,2,2]
=> [5,1,1,1,1,1,1,1,1]
=> ? = 3
[3,3,2,2,2,1,1,1,1]
=> [3,2,2,2,1,1,1,1]
=> [8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 3
[3,3,2,2,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1]
=> [9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> ? = 3
[3,3,2,1,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [10,2,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 3
[3,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2
[3,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> ? = 2
[3,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> ? = 2
[3,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 2
[2,2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,2,1,1]
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 2
[2,2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1]
=> [9,5]
=> [2,2,2,2,2,1,1,1,1]
=> ? = 2
[2,2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1,1]
=> [10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> ? = 2
[2,2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [11,3]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> ? = 2
[2,2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 2
[4,4,4,4,1]
=> [4,4,4,1]
=> [4,3,3,3]
=> [7,1,1,1,1,1,1]
=> ? = 4
[4,4,4,2,2,1]
=> [4,4,2,2,1]
=> [5,4,2,2]
=> [9,4]
=> ? = 4
[4,4,4,2,1,1,1]
=> [4,4,2,1,1,1]
=> [6,3,2,2]
=> [3,3,3,2,1,1]
=> ? = 4
[4,4,4,1,1,1,1,1]
=> [4,4,1,1,1,1,1]
=> [7,2,2,2]
=> [3,3,3,1,1,1,1]
=> ? = 4
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[4,4,3,3,1,1,1]
=> [4,3,3,1,1,1]
=> [6,3,3,1]
=> [8,2,1,1,1]
=> ? = 4
[4,4,3,2,2,1,1]
=> [4,3,2,2,1,1]
=> [6,4,2,1]
=> [7,2,2,1,1]
=> ? = 4
[4,4,3,2,1,1,1,1]
=> [4,3,2,1,1,1,1]
=> [7,3,2,1]
=> [6,2,1,1,1,1,1]
=> ? = 4
[4,4,2,2,2,2,1]
=> [4,2,2,2,2,1]
=> [6,5,1,1]
=> [11,2]
=> ? = 4
[4,4,2,2,2,1,1,1]
=> [4,2,2,2,1,1,1]
=> [7,4,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 4
[4,4,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> [9,2,1,1]
=> [3,3,1,1,1,1,1,1,1]
=> ? = 4
[4,4,1,1,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> [10,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> ? = 4
[4,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> [10,1,1,1]
=> ? = 3
[4,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 3
[4,3,3,3,1,1,1,1]
=> [3,3,3,1,1,1,1]
=> [7,3,3]
=> [8,1,1,1,1,1]
=> ? = 3
[4,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> [11,1,1]
=> ? = 3
[4,3,3,2,2,1,1,1]
=> [3,3,2,2,1,1,1]
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3
[4,3,3,1,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1,1]
=> [9,2,2]
=> [5,1,1,1,1,1,1,1,1]
=> ? = 3
[4,3,2,2,2,1,1,1,1]
=> [3,2,2,2,1,1,1,1]
=> [8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 3
[4,3,2,2,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1]
=> [9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> ? = 3
[4,3,2,1,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [10,2,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 3
[4,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2
[4,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> ? = 2
[4,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> ? = 2
[4,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 2
[3,3,3,3,3,1,1]
=> [3,3,3,3,1,1]
=> [6,4,4]
=> [10,1,1,1,1]
=> ? = 3
[3,3,3,3,2,2,1]
=> [3,3,3,2,2,1]
=> [6,5,3]
=> [11,1,1,1]
=> ? = 3
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St001280
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 86%●distinct values known / distinct values provided: 69%
St001280: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 86%●distinct values known / distinct values provided: 69%
Values
[1]
=> [1]
=> 0
[2]
=> [1,1]
=> 0
[1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> 0
[2,1]
=> [2,1]
=> 1
[1,1,1]
=> [3]
=> 1
[4]
=> [1,1,1,1]
=> 0
[3,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> 2
[2,1,1]
=> [3,1]
=> 1
[1,1,1,1]
=> [4]
=> 1
[5]
=> [1,1,1,1,1]
=> 0
[4,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> 2
[3,1,1]
=> [3,1,1]
=> 1
[2,2,1]
=> [3,2]
=> 2
[2,1,1,1]
=> [4,1]
=> 1
[1,1,1,1,1]
=> [5]
=> 1
[6]
=> [1,1,1,1,1,1]
=> 0
[5,1]
=> [2,1,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> 1
[3,3]
=> [2,2,2]
=> 3
[3,2,1]
=> [3,2,1]
=> 2
[3,1,1,1]
=> [4,1,1]
=> 1
[2,2,2]
=> [3,3]
=> 2
[2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> 1
[1,1,1,1,1,1]
=> [6]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> 1
[4,3]
=> [2,2,2,1]
=> 3
[4,2,1]
=> [3,2,1,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> 1
[3,3,1]
=> [3,2,2]
=> 3
[3,2,2]
=> [3,3,1]
=> 2
[3,2,1,1]
=> [4,2,1]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> 1
[2,2,2,1]
=> [4,3]
=> 2
[2,2,1,1,1]
=> [5,2]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> 1
[5,3]
=> [2,2,2,1,1]
=> 3
[5,2,1]
=> [3,2,1,1,1]
=> 2
[17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[16,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[15,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 2
[15,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[14,3]
=> [2,2,2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3
[14,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 2
[13,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1,1]
=> ? = 4
[12,5]
=> [2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 5
[11,6]
=> [2,2,2,2,2,2,1,1,1,1,1]
=> ? = 6
[10,7]
=> [2,2,2,2,2,2,2,1,1,1]
=> ? = 7
[9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 8
[6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 5
[5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? = 5
[6,4,4,3,2,1]
=> [6,5,4,3,1,1]
=> ? = 4
[5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? = 4
[4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 4
[6,5,3,3,2,1]
=> [6,5,4,2,2,1]
=> ? = 5
[5,5,3,3,2,1]
=> [6,5,4,2,2]
=> ? = 5
[6,4,3,3,2,1]
=> [6,5,4,2,1,1]
=> ? = 4
[5,4,3,3,2,1]
=> [6,5,4,2,1]
=> ? = 4
[6,3,3,3,2,1]
=> [6,5,4,1,1,1]
=> ? = 3
[6,5,4,2,2,1]
=> [6,5,3,3,2,1]
=> ? = 5
[5,5,4,2,2,1]
=> [6,5,3,3,2]
=> ? = 5
[6,4,4,2,2,1]
=> [6,5,3,3,1,1]
=> ? = 4
[5,4,4,2,2,1]
=> [6,5,3,3,1]
=> ? = 4
[6,5,3,2,2,1]
=> [6,5,3,2,2,1]
=> ? = 5
[5,5,3,2,2,1]
=> [6,5,3,2,2]
=> ? = 5
[6,4,3,2,2,1]
=> [6,5,3,2,1,1]
=> ? = 4
[6,5,2,2,2,1]
=> [6,5,2,2,2,1]
=> ? = 5
[6,5,4,3,1,1]
=> [6,4,4,3,2,1]
=> ? = 5
[5,5,4,3,1,1]
=> [6,4,4,3,2]
=> ? = 5
[6,4,4,3,1,1]
=> [6,4,4,3,1,1]
=> ? = 4
[5,4,4,3,1,1]
=> [6,4,4,3,1]
=> ? = 4
[6,5,3,3,1,1]
=> [6,4,4,2,2,1]
=> ? = 5
[5,5,3,3,1,1]
=> [6,4,4,2,2]
=> ? = 5
[6,4,3,3,1,1]
=> [6,4,4,2,1,1]
=> ? = 4
[6,5,4,2,1,1]
=> [6,4,3,3,2,1]
=> ? = 5
[5,5,4,2,1,1]
=> [6,4,3,3,2]
=> ? = 5
[6,4,4,2,1,1]
=> [6,4,3,3,1,1]
=> ? = 4
[6,5,3,2,1,1]
=> [6,4,3,2,2,1]
=> ? = 5
[6,5,4,1,1,1]
=> [6,3,3,3,2,1]
=> ? = 5
[6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 5
[5,5,4,3,2]
=> [5,5,4,3,2]
=> ? = 5
[6,4,4,3,2]
=> [5,5,4,3,1,1]
=> ? = 4
[5,4,4,3,2]
=> [5,5,4,3,1]
=> ? = 4
[6,5,3,3,2]
=> [5,5,4,2,2,1]
=> ? = 5
[5,5,3,3,2]
=> [5,5,4,2,2]
=> ? = 5
[6,4,3,3,2]
=> [5,5,4,2,1,1]
=> ? = 4
[6,5,4,2,2]
=> [5,5,3,3,2,1]
=> ? = 5
[5,5,4,2,2]
=> [5,5,3,3,2]
=> ? = 5
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000476
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 83%●distinct values known / distinct values provided: 54%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 83%●distinct values known / distinct values provided: 54%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[5]
=> []
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[6]
=> []
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[7]
=> []
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[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]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[8]
=> []
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[9]
=> []
=> []
=> []
=> ? = 0
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[10]
=> []
=> []
=> []
=> ? = 0
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[11]
=> []
=> []
=> []
=> ? = 0
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[12]
=> []
=> []
=> []
=> ? = 0
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[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]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 1
[13]
=> []
=> []
=> []
=> ? = 0
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[3,2,2,1,1,1,1,1,1]
=> [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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[3,2,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[3,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[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]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[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]
=> [1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 2
[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]
=> ?
=> ? = 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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 1
[14]
=> []
=> []
=> []
=> ? = 0
[7,7]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[6,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,2,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[5,1,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[4,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[4,2,2,1,1,1,1,1,1]
=> [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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[4,2,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[4,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[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]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[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]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 79%●distinct values known / distinct values provided: 77%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 79%●distinct values known / distinct values provided: 77%
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 => 1
[4]
=> []
=> []
=> => ? = 0
[3,1]
=> [1]
=> [1]
=> 10 => 1
[2,2]
=> [2]
=> [1,1]
=> 110 => 2
[2,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[5]
=> []
=> []
=> => ? = 0
[4,1]
=> [1]
=> [1]
=> 10 => 1
[3,2]
=> [2]
=> [1,1]
=> 110 => 2
[3,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[2,2,1]
=> [2,1]
=> [2,1]
=> 1010 => 2
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 10000 => 1
[6]
=> []
=> []
=> => ? = 0
[5,1]
=> [1]
=> [1]
=> 10 => 1
[4,2]
=> [2]
=> [1,1]
=> 110 => 2
[4,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[3,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[3,2,1]
=> [2,1]
=> [2,1]
=> 1010 => 2
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[2,2,2]
=> [2,2]
=> [2,2]
=> 1100 => 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 10010 => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 10000 => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 100000 => 1
[7]
=> []
=> []
=> => ? = 0
[6,1]
=> [1]
=> [1]
=> 10 => 1
[5,2]
=> [2]
=> [1,1]
=> 110 => 2
[5,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[4,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[4,2,1]
=> [2,1]
=> [2,1]
=> 1010 => 2
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 10110 => 3
[3,2,2]
=> [2,2]
=> [2,2]
=> 1100 => 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 10010 => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 10000 => 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 10100 => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 100010 => 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 100000 => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1000000 => 1
[8]
=> []
=> []
=> => ? = 0
[7,1]
=> [1]
=> [1]
=> 10 => 1
[6,2]
=> [2]
=> [1,1]
=> 110 => 2
[6,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[5,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[5,2,1]
=> [2,1]
=> [2,1]
=> 1010 => 2
[5,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[4,4]
=> [4]
=> [1,1,1,1]
=> 11110 => 4
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 10110 => 3
[4,2,2]
=> [2,2]
=> [2,2]
=> 1100 => 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 10010 => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 10000 => 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> 11010 => 3
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
[9]
=> []
=> []
=> => ? = 0
[10]
=> []
=> []
=> => ? = 0
[11]
=> []
=> []
=> => ? = 0
[12]
=> []
=> []
=> => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> 100000000000 => ? = 1
[13]
=> []
=> []
=> => ? = 0
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> 1000001000 => ? = 2
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? = 2
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> 100000000000 => ? = 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]
=> [12]
=> 1000000000000 => ? = 1
[14]
=> []
=> []
=> => ? = 0
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> 1000001100 => ? = 3
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> 1000010010 => ? = 3
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> 100000000110 => ? = 3
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> 1000001000 => ? = 2
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? = 2
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> 100000000000 => ? = 1
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> 1000010000 => ? = 2
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [9,3]
=> 10000001000 => ? = 2
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [10,2]
=> 100000000100 => ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [11,1]
=> 1000000000010 => ? = 2
[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]
=> [12]
=> 1000000000000 => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> 10000000000000 => ? = 1
[15]
=> []
=> []
=> => ? = 0
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [6,2,2,1]
=> 1000011010 => ? = 4
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [6,3,1,1]
=> 1000100110 => ? = 4
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> 100000001110 => ? = 4
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> 1000001100 => ? = 3
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> 1000010010 => ? = 3
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> 100000000110 => ? = 3
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> 1000001000 => ? = 2
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? = 2
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> 100000000000 => ? = 1
[3,3,3,3,1,1,1]
=> [3,3,3,1,1,1]
=> [6,3,3]
=> 100011000 => ? = 3
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [7,3,2]
=> 1000010100 => ? = 3
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> 10000001100 => ? = 3
[3,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [6,5,1]
=> 101000010 => ? = 3
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [7,4,1]
=> 1000100010 => ? = 3
[3,3,2,2,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1]
=> [8,3,1]
=> 10000010010 => ? = 3
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [9,2,1]
=> 100000001010 => ? = 3
[3,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [10,1,1]
=> 1000000000110 => ? = 3
[3,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> 1000010000 => ? = 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000013
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 85%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 85%
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]
=> 1
[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]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[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]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[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]
=> 1
[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]
=> 1
[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]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[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]
=> 1
[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]
=> 1
[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]
=> 1
[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]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[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]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[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]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[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]
=> 1
[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]
=> 2
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 1
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 1
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3
[3,3,2,2,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 3
[3,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 3
[3,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[3,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 1
[2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 2
[2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 1
[5,5,3,2,1]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[5,5,3,1,1,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 5
[5,5,2,2,1,1]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[5,5,2,1,1,1,1]
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[5,5,1,1,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[5,4,4,1,1,1]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[5,4,3,2,1,1]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[5,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[5,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[5,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[5,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[5,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3
[5,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[5,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[5,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[4,4,4,2,1,1]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 4
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000734
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 77%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 77%
Values
[1]
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [[1]]
=> 1
[3]
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [[1]]
=> 1
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[4]
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [[1]]
=> 1
[2,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[5]
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [[1]]
=> 1
[3,2]
=> [2]
=> [[1,2]]
=> 2
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[6]
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [[1]]
=> 1
[4,2]
=> [2]
=> [[1,2]]
=> 2
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[3,3]
=> [3]
=> [[1,2,3]]
=> 3
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[7]
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [[1]]
=> 1
[5,2]
=> [2]
=> [[1,2]]
=> 2
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[4,3]
=> [3]
=> [[1,2,3]]
=> 3
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[8]
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [[1]]
=> 1
[6,2]
=> [2]
=> [[1,2]]
=> 2
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[5,3]
=> [3]
=> [[1,2,3]]
=> 3
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> 4
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[9]
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> ? = 0
[11]
=> []
=> []
=> ? = 0
[12]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[13]
=> []
=> []
=> ? = 0
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 2
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 1
[14]
=> []
=> []
=> ? = 0
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? = 3
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? = 3
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 3
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? = 3
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 2
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 2
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 1
[15]
=> []
=> []
=> ? = 0
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? = 4
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 4
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 4
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 4
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? = 3
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? = 3
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 3
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? = 3
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000054
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 77%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 77%
Values
[1]
=> []
=> []
=> [] => ? = 0 + 1
[2]
=> []
=> []
=> [] => ? = 0 + 1
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[3]
=> []
=> []
=> [] => ? = 0 + 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[4]
=> []
=> []
=> [] => ? = 0 + 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[5]
=> []
=> []
=> [] => ? = 0 + 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 2 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[6]
=> []
=> []
=> [] => ? = 0 + 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 2 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3 = 2 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[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] => 2 = 1 + 1
[7]
=> []
=> []
=> [] => ? = 0 + 1
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 2 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 3 + 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3 = 2 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3 = 2 + 1
[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] => 2 = 1 + 1
[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] => 2 = 1 + 1
[8]
=> []
=> []
=> [] => ? = 0 + 1
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 2 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 5 = 4 + 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 3 + 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3 = 2 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 4 = 3 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4 = 3 + 1
[9]
=> []
=> []
=> [] => ? = 0 + 1
[10]
=> []
=> []
=> [] => ? = 0 + 1
[11]
=> []
=> []
=> [] => ? = 0 + 1
[12]
=> []
=> []
=> [] => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [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 + 1
[13]
=> []
=> []
=> [] => ? = 0 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 1 + 1
[14]
=> []
=> []
=> [] => ? = 0 + 1
[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] => ? = 3 + 1
[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] => ? = 3 + 1
[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] => ? = 3 + 1
[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] => ? = 3 + 1
[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] => ? = 3 + 1
[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] => ? = 3 + 1
[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] => ? = 3 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[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] => ? = 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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 1 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[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] => ? = 2 + 1
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,12,1] => ? = 2 + 1
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 1 + 1
[15]
=> []
=> []
=> [] => ? = 0 + 1
[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] => ? = 4 + 1
[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] => ? = 4 + 1
[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] => ? = 4 + 1
[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] => ? = 4 + 1
[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] => ? = 3 + 1
[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] => ? = 3 + 1
[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] => ? = 3 + 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000141
(load all 2 compositions to match this statistic)
(load all 2 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
St000141: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 77%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 77%
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] => 1
[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] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[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] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[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] => 1
[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] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[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] => 1
[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] => 1
[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] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[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] => 1
[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] => 1
[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] => 1
[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] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[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] => 3
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[9]
=> []
=> []
=> [] => ? = 0
[10]
=> []
=> []
=> [] => ? = 0
[11]
=> []
=> []
=> [] => ? = 0
[12]
=> []
=> []
=> [] => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [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
[13]
=> []
=> []
=> [] => ? = 0
[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] => ? = 2
[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] => ? = 2
[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] => ? = 2
[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] => ? = 2
[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] => ? = 2
[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]
=> ? => ? = 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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 1
[14]
=> []
=> []
=> [] => ? = 0
[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] => ? = 3
[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] => ? = 3
[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] => ? = 3
[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] => ? = 3
[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] => ? = 3
[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] => ? = 3
[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] => ? = 3
[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] => ? = 2
[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] => ? = 2
[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] => ? = 2
[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] => ? = 2
[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] => ? = 2
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 1
[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] => ? = 2
[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] => ? = 2
[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] => ? = 2
[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] => ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,12,1] => ? = 2
[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]
=> ? => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 1
[15]
=> []
=> []
=> [] => ? = 0
[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] => ? = 4
[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] => ? = 4
[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] => ? = 4
[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] => ? = 4
[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] => ? = 3
[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] => ? = 3
[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] => ? = 3
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
The following 82 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000733The row containing the largest entry of a standard tableau. St000691The number of changes of a binary word. St000676The number of odd rises of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000439The position of the first down step of a Dyck path. St001497The position of the largest weak excedence of a permutation. St000653The last descent of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000025The number of initial rises of a Dyck path. St000444The length of the maximal rise of 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. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St000381The largest part of an integer composition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. 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. St001480The number of simple summands of the module J^2/J^3. St000808The number of up steps of the associated bargraph. St000031The number of cycles in the cycle decomposition of a permutation. St000738The first entry in the last row of a standard tableau. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St000383The last part of an integer composition. St000443The number of long tunnels of a Dyck path. 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. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000083The number of left oriented leafs of a binary tree except the first one. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000840The number of closers smaller than the largest opener in a perfect matching. St000504The cardinality of the first block of a set partition. St001062The maximal size of a block of a set partition. St000823The number of unsplittable factors of the set partition. St000503The maximal difference between two elements in a common block. St000740The last entry of a permutation. 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. St000746The number of pairs with odd minimum in a perfect matching. St000308The height of the tree associated to a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000989The number of final rises of a permutation. St000157The number of descents of a standard tableau. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000451The length of the longest pattern of the form k 1 2. St000028The number of stack-sorts needed to sort a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000651The maximal size of a rise in a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000742The number of big ascents of a permutation after prepending zero. St000035The number of left outer peaks of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000730The maximal arc length of a set partition. St001674The number of vertices of the largest induced star graph in the graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001737The number of descents of type 2 in a permutation. St001665The number of pure excedances of a permutation. St001928The number of non-overlapping descents in a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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