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Your data matches 138 different statistics following compositions of up to 3 maps.
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Matching statistic: St000734
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1]
=> [1]
=> [[1]]
=> 1
[3]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [[1]]
=> 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [[1]]
=> 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> [[1,2,3]]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [[1]]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [[1,2,3]]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [[1]]
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000147
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
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%
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%
Values
[2]
=> [1,1]
=> [1]
=> 1
[3]
=> [1,1,1]
=> [1,1]
=> 1
[2,1]
=> [2,1]
=> [1]
=> 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2]
=> [2,2]
=> [2]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> 2
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> 2
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 3
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> 3
[10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> ? = 2
[7,7,7]
=> [3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> ? = 3
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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%
Values
[2]
=> [1,1]
=> [1]
=> [1]
=> 1
[3]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [1]
=> 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [1]
=> 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1]
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [3,2]
=> 2
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 3
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [2,2,1]
=> 3
[10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 2
[7,7,7]
=> [3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [6,6,6]
=> ? = 3
Description
The length of the partition.
Matching statistic: St001280
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
St001280: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[2]
=> []
=> ? = 1 - 1
[3]
=> []
=> ? = 1 - 1
[2,1]
=> [1]
=> 0 = 1 - 1
[4]
=> []
=> ? = 1 - 1
[3,1]
=> [1]
=> 0 = 1 - 1
[2,2]
=> [2]
=> 1 = 2 - 1
[2,1,1]
=> [1,1]
=> 0 = 1 - 1
[5]
=> []
=> ? = 1 - 1
[4,1]
=> [1]
=> 0 = 1 - 1
[3,2]
=> [2]
=> 1 = 2 - 1
[3,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[6]
=> []
=> ? = 1 - 1
[5,1]
=> [1]
=> 0 = 1 - 1
[4,2]
=> [2]
=> 1 = 2 - 1
[4,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,3]
=> [3]
=> 1 = 2 - 1
[3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,2,2]
=> [2,2]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[7]
=> []
=> ? = 1 - 1
[6,1]
=> [1]
=> 0 = 1 - 1
[5,2]
=> [2]
=> 1 = 2 - 1
[5,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,3]
=> [3]
=> 1 = 2 - 1
[4,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[3,3,1]
=> [3,1]
=> 1 = 2 - 1
[3,2,2]
=> [2,2]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> 1 = 2 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[2,2,2,1]
=> [2,2,1]
=> 2 = 3 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[8]
=> []
=> ? = 1 - 1
[7,1]
=> [1]
=> 0 = 1 - 1
[6,2]
=> [2]
=> 1 = 2 - 1
[6,1,1]
=> [1,1]
=> 0 = 1 - 1
[5,3]
=> [3]
=> 1 = 2 - 1
[5,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,4]
=> [4]
=> 1 = 2 - 1
[4,3,1]
=> [3,1]
=> 1 = 2 - 1
[4,2,2]
=> [2,2]
=> 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> 1 = 2 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[3,3,2]
=> [3,2]
=> 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> 1 = 2 - 1
[3,2,2,1]
=> [2,2,1]
=> 2 = 3 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[2,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> 2 = 3 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[9]
=> []
=> ? = 1 - 1
[10]
=> []
=> ? = 1 - 1
[11]
=> []
=> ? = 1 - 1
[5,5,5,1,1,1,1]
=> ?
=> ? = 3 - 1
[3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> ? = 7 - 1
[3,3,3,3,2,2,1,1,1,1]
=> ?
=> ? = 6 - 1
[4,4,3,3,2,2,1,1]
=> ?
=> ? = 6 - 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000676
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 89%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 89%
Values
[2]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0]
=> 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0]
=> 1
[7]
=> [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
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0]
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10]
=> [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
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[11]
=> [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
[10,1]
=> [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
[9,2]
=> [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]
=> ? = 2
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,3]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,2]
=> [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]
=> ? = 2
[10,1,1]
=> [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
[9,3]
=> [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]
=> ? = 2
[9,2,1]
=> [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]
=> ? = 2
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4]
=> [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]
=> ? = 2
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,2,2]
=> [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]
=> ? = 3
[11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,2,1]
=> [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]
=> ? = 2
[10,1,1,1]
=> [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
[9,3,1]
=> [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]
=> ? = 2
[9,2,2]
=> [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]
=> ? = 3
[9,2,1,1]
=> [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]
=> ? = 2
[9,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4,1]
=> [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]
=> ? = 2
[8,3,2]
=> [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]
=> ? = 3
[8,3,1,1]
=> [4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,2,2,1]
=> [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]
=> ? = 3
[7,3,3]
=> [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]
=> ? = 3
[7,2,2,2]
=> [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]
=> ? = 4
[11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,2,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 2
[10,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1,1]
=> [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
[9,3,1,1]
=> [4,2,2,1,1,1,1,1,1]
=> [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]
=> ? = 2
[9,2,2,1]
=> [4,3,1,1,1,1,1,1,1]
=> [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]
=> ? = 3
[9,2,1,1,1]
=> [5,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 2
[9,1,1,1,1,1]
=> [6,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4,1,1]
=> [4,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,3,2,1]
=> [4,3,2,1,1,1,1,1]
=> [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]
=> ? = 3
[8,3,1,1,1]
=> [5,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,2,2,2]
=> [4,4,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[8,2,2,1,1]
=> [5,3,1,1,1,1,1,1]
=> [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]
=> ? = 3
[7,3,3,1]
=> [4,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[7,3,2,2]
=> [4,4,2,1,1,1,1]
=> [4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[7,2,2,2,1]
=> [5,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[6,2,2,2,2]
=> [5,5,1,1,1,1]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[11,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1,1,1]
=> [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
[10,2,1,1,1]
=> [5,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 2
[10,1,1,1,1,1]
=> [6,1,1,1,1,1,1,1,1,1]
=> [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
[9,3,1,1,1]
=> [5,2,2,1,1,1,1,1,1]
=> [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]
=> ? = 2
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001039
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 89%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 89%
Values
[2]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
[7]
=> [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
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,3,1,1]
=> [4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,2,1]
=> [4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,1,1,1]
=> [5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1,1,1]
=> [6,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2,2]
=> [4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,2,2,1,1]
=> [5,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [1]
=> [1,0]
=> ? = 1
[10]
=> [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
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1]
=> [1,0]
=> ? = 1
[11]
=> [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
[10,1]
=> [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
[9,2]
=> [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]
=> ? = 2
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,3]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [1]
=> [1,0]
=> ? = 1
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,2]
=> [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]
=> ? = 2
[10,1,1]
=> [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
[9,3]
=> [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]
=> ? = 2
[9,2,1]
=> [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]
=> ? = 2
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4]
=> [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]
=> ? = 2
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,2,2]
=> [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]
=> ? = 3
[2,1,1,1,1,1,1,1,1,1,1]
=> [11,1]
=> [1]
=> [1,0]
=> ? = 1
[11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,2,1]
=> [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]
=> ? = 2
[10,1,1,1]
=> [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
[9,3,1]
=> [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]
=> ? = 2
[9,2,2]
=> [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]
=> ? = 3
[9,2,1,1]
=> [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]
=> ? = 2
[9,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4,1]
=> [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]
=> ? = 2
[8,3,2]
=> [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]
=> ? = 3
[8,3,1,1]
=> [4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,2,2,1]
=> [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]
=> ? = 3
[7,3,3]
=> [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]
=> ? = 3
[7,2,2,2]
=> [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]
=> ? = 4
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [12,1]
=> [1]
=> [1,0]
=> ? = 1
[11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,2,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 2
[10,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1,1]
=> [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
[9,3,1,1]
=> [4,2,2,1,1,1,1,1,1]
=> [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]
=> ? = 2
[9,2,2,1]
=> [4,3,1,1,1,1,1,1,1]
=> [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]
=> ? = 3
[9,2,1,1,1]
=> [5,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 2
[9,1,1,1,1,1]
=> [6,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4,1,1]
=> [4,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 72%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 72%●distinct values known / distinct values provided: 56%
Values
[2]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[7]
=> [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 + 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[8]
=> [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]
=> ? = 1 + 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[3,3,1,1]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,2,1]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,2,1,1,1]
=> [5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[9]
=> [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 + 1
[8,1]
=> [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]
=> ? = 1 + 1
[7,2]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7,1,1]
=> [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[10]
=> [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 + 1
[9,1]
=> [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 + 1
[8,2]
=> [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]
=> ? = 2 + 1
[8,1,1]
=> [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]
=> ? = 1 + 1
[7,3]
=> [2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,2,1]
=> [3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7,1,1,1]
=> [4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[11]
=> [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 + 1
[10,1]
=> [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 + 1
[9,2]
=> [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]
=> ? = 2 + 1
[9,1,1]
=> [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 + 1
[8,3]
=> [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]
=> ? = 2 + 1
[8,2,1]
=> [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]
=> ? = 2 + 1
[8,1,1,1]
=> [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]
=> ? = 1 + 1
[7,4]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,3,1]
=> [3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,2,2]
=> [3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[11,1]
=> [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 + 1
[10,2]
=> [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]
=> ? = 2 + 1
[10,1,1]
=> [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 + 1
[9,3]
=> [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]
=> ? = 2 + 1
[9,2,1]
=> [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]
=> ? = 2 + 1
[9,1,1,1]
=> [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 + 1
[8,4]
=> [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]
=> ? = 2 + 1
[8,3,1]
=> [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]
=> ? = 2 + 1
[8,2,2]
=> [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]
=> ? = 3 + 1
[8,2,1,1]
=> [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]
=> ? = 2 + 1
[8,1,1,1,1]
=> [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]
=> ? = 1 + 1
[7,5]
=> [2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,4,1]
=> [3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,3,2]
=> [3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 1
[7,3,1,1]
=> [4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,2,2,1]
=> [4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[7,2,1,1,1]
=> [5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7,1,1,1,1,1]
=> [6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[6,6]
=> [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,2,2,2,2,2]
=> [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[10,2,1]
=> [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]
=> ? = 2 + 1
[10,1,1,1]
=> [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 + 1
[9,3,1]
=> [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]
=> ? = 2 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000665
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
[2]
=> []
=> []
=> [] => 0 = 1 - 1
[3]
=> []
=> []
=> [] => 0 = 1 - 1
[2,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[4]
=> []
=> []
=> [] => 0 = 1 - 1
[3,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[5]
=> []
=> []
=> [] => 0 = 1 - 1
[4,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[6]
=> []
=> []
=> [] => 0 = 1 - 1
[5,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 2 - 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[7]
=> []
=> []
=> [] => 0 = 1 - 1
[6,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 2 - 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 2 - 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 2 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 3 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 2 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[8]
=> []
=> []
=> [] => 0 = 1 - 1
[7,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 2 - 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 2 - 1
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 2 - 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 2 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2 = 3 - 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 6 - 1
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? = 5 - 1
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 4 - 1
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1 - 1
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 5 - 1
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? = 4 - 1
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 5 - 1
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? = 4 - 1
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? = 6 - 1
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? = 5 - 1
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? = 4 - 1
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 2 - 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 6 - 1
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? = 5 - 1
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 4 - 1
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1 - 1
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 7 - 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]]
=> ? => ? = 6 - 1
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? = 5 - 1
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 4 - 1
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 3 - 1
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 2 - 1
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 4 - 1
[4,4,4,2,1]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [11,9,10,5,6,7,8,1,2,3,4] => ? = 4 - 1
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [11,10,9,5,6,7,8,1,2,3,4] => ? = 3 - 1
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 4 - 1
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [10,11,8,9,5,6,7,1,2,3,4] => ? = 5 - 1
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 4 - 1
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,5,6,1,2,3,4] => ? = 5 - 1
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? = 4 - 1
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 2 - 1
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 5 - 1
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? = 4 - 1
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 5 - 1
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? = 4 - 1
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? = 6 - 1
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? = 5 - 1
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? = 4 - 1
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 2 - 1
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 6 - 1
Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Matching statistic: St000834
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
[2]
=> []
=> []
=> [] => 0 = 1 - 1
[3]
=> []
=> []
=> [] => 0 = 1 - 1
[2,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[4]
=> []
=> []
=> [] => 0 = 1 - 1
[3,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[5]
=> []
=> []
=> [] => 0 = 1 - 1
[4,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[6]
=> []
=> []
=> [] => 0 = 1 - 1
[5,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 2 - 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[7]
=> []
=> []
=> [] => 0 = 1 - 1
[6,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 2 - 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 2 - 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 2 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 3 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 2 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[8]
=> []
=> []
=> [] => 0 = 1 - 1
[7,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1 = 2 - 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 2 - 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 2 - 1
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 2 - 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 2 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2 = 3 - 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 6 - 1
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? = 5 - 1
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 4 - 1
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1 - 1
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 5 - 1
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? = 4 - 1
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 5 - 1
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? = 4 - 1
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? = 6 - 1
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? = 5 - 1
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? = 4 - 1
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 2 - 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 6 - 1
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? = 5 - 1
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 4 - 1
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1 - 1
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 7 - 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]]
=> ? => ? = 6 - 1
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? = 5 - 1
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 4 - 1
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 3 - 1
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 2 - 1
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 4 - 1
[4,4,4,2,1]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [11,9,10,5,6,7,8,1,2,3,4] => ? = 4 - 1
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [11,10,9,5,6,7,8,1,2,3,4] => ? = 3 - 1
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 4 - 1
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [10,11,8,9,5,6,7,1,2,3,4] => ? = 5 - 1
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 4 - 1
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,5,6,1,2,3,4] => ? = 5 - 1
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? = 4 - 1
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 2 - 1
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 5 - 1
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? = 4 - 1
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 5 - 1
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? = 4 - 1
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? = 6 - 1
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? = 5 - 1
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? = 4 - 1
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 3 - 1
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 2 - 1
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 6 - 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000378
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 78%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 78%
Values
[2]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[3]
=> [2,1]
=> [3]
=> 2 = 1 + 1
[2,1]
=> [2,1]
=> [3]
=> 2 = 1 + 1
[4]
=> [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[3,1]
=> [2,2]
=> [4]
=> 2 = 1 + 1
[2,2]
=> [2,1,1]
=> [2,2]
=> 3 = 2 + 1
[2,1,1]
=> [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[5]
=> [4,1]
=> [2,1,1,1]
=> 2 = 1 + 1
[4,1]
=> [3,2]
=> [5]
=> 2 = 1 + 1
[3,2]
=> [2,2,1]
=> [2,2,1]
=> 3 = 2 + 1
[3,1,1]
=> [3,2]
=> [5]
=> 2 = 1 + 1
[2,2,1]
=> [3,1,1]
=> [4,1]
=> 3 = 2 + 1
[2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2 = 1 + 1
[6]
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 1 + 1
[5,1]
=> [4,2]
=> [2,2,1,1]
=> 2 = 1 + 1
[4,2]
=> [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[4,1,1]
=> [3,3]
=> [6]
=> 2 = 1 + 1
[3,3]
=> [2,2,2]
=> [2,2,2]
=> 3 = 2 + 1
[3,2,1]
=> [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[3,1,1,1]
=> [4,2]
=> [2,2,1,1]
=> 2 = 1 + 1
[2,2,2]
=> [3,1,1,1]
=> [3,3]
=> 4 = 3 + 1
[2,2,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 1 + 1
[7]
=> [6,1]
=> [2,1,1,1,1,1]
=> 2 = 1 + 1
[6,1]
=> [5,2]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[5,2]
=> [4,2,1]
=> [5,1,1]
=> 3 = 2 + 1
[5,1,1]
=> [4,3]
=> [7]
=> 2 = 1 + 1
[4,3]
=> [3,2,2]
=> [2,2,2,1]
=> 3 = 2 + 1
[4,2,1]
=> [3,3,1]
=> [6,1]
=> 3 = 2 + 1
[4,1,1,1]
=> [4,3]
=> [7]
=> 2 = 1 + 1
[3,3,1]
=> [3,2,2]
=> [2,2,2,1]
=> 3 = 2 + 1
[3,2,2]
=> [3,2,1,1]
=> [5,2]
=> 4 = 3 + 1
[3,2,1,1]
=> [4,2,1]
=> [5,1,1]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[2,2,2,1]
=> [4,1,1,1]
=> [3,2,1,1]
=> 4 = 3 + 1
[2,2,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> 2 = 1 + 1
[8]
=> [7,1]
=> [2,1,1,1,1,1,1]
=> 2 = 1 + 1
[7,1]
=> [6,2]
=> [2,2,1,1,1,1]
=> 2 = 1 + 1
[6,2]
=> [5,2,1]
=> [4,1,1,1,1]
=> 3 = 2 + 1
[6,1,1]
=> [5,3]
=> [2,2,2,1,1]
=> 2 = 1 + 1
[5,3]
=> [4,2,2]
=> [6,1,1]
=> 3 = 2 + 1
[5,2,1]
=> [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[5,1,1,1]
=> [4,4]
=> [8]
=> 2 = 1 + 1
[4,4]
=> [3,3,2]
=> [2,2,2,2]
=> 3 = 2 + 1
[4,3,1]
=> [3,3,2]
=> [2,2,2,2]
=> 3 = 2 + 1
[4,2,2]
=> [3,3,1,1]
=> [6,2]
=> 4 = 3 + 1
[4,2,1,1]
=> [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[4,1,1,1,1]
=> [5,3]
=> [2,2,2,1,1]
=> 2 = 1 + 1
[3,3,2]
=> [3,2,2,1]
=> [3,2,2,1]
=> 4 = 3 + 1
[11,1,1]
=> [10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> ? = 1 + 1
[10,2,1]
=> [9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> ? = 2 + 1
[9,2,1,1]
=> [8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 2 + 1
[9,1,1,1,1]
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 1 + 1
[8,4,1]
=> [7,3,3]
=> [8,1,1,1,1,1]
=> ? = 2 + 1
[8,3,2]
=> [7,3,2,1]
=> [6,2,1,1,1,1,1]
=> ? = 3 + 1
[8,3,1,1]
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 2 + 1
[8,2,2,1]
=> [7,4,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 3 + 1
[7,6]
=> [6,5,2]
=> [11,1,1]
=> ? = 2 + 1
[7,5,1]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 2 + 1
[7,4,2]
=> [6,3,3,1]
=> [8,2,1,1,1]
=> ? = 3 + 1
[7,4,1,1]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 2 + 1
[7,3,3]
=> [6,3,2,2]
=> [3,3,3,2,1,1]
=> ? = 3 + 1
[7,3,2,1]
=> [6,4,2,1]
=> [7,2,2,1,1]
=> ? = 3 + 1
[7,3,1,1,1]
=> [6,5,2]
=> [11,1,1]
=> ? = 2 + 1
[7,2,2,2]
=> [6,4,1,1,1]
=> [4,3,2,2,1,1]
=> ? = 4 + 1
[7,2,2,1,1]
=> [6,5,1,1]
=> [11,2]
=> ? = 3 + 1
[6,6,1]
=> [5,5,3]
=> [10,1,1,1]
=> ? = 2 + 1
[6,5,2]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 3 + 1
[6,4,2,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 3 + 1
[6,4,1,1,1]
=> [5,5,3]
=> [10,1,1,1]
=> ? = 2 + 1
[6,3,3,1]
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
[6,3,1,1,1,1]
=> [6,5,2]
=> [11,1,1]
=> ? = 2 + 1
[6,2,2,2,1]
=> [5,5,1,1,1]
=> [10,2,1]
=> ? = 4 + 1
[6,2,2,1,1,1]
=> [6,5,1,1]
=> [11,2]
=> ? = 3 + 1
[6,1,1,1,1,1,1,1]
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 1 + 1
[5,4,4]
=> [4,3,3,3]
=> [7,1,1,1,1,1,1]
=> ? = 3 + 1
[5,4,2,2]
=> [4,4,3,1,1]
=> [4,2,2,2,2,1]
=> ? = 4 + 1
[5,4,2,1,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 3 + 1
[5,4,1,1,1,1]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 2 + 1
[5,3,3,1,1]
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
[5,3,2,1,1,1]
=> [6,4,2,1]
=> [7,2,2,1,1]
=> ? = 3 + 1
[5,3,1,1,1,1,1]
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 2 + 1
[5,2,2,2,2]
=> [5,4,1,1,1,1]
=> [9,3,1]
=> ? = 5 + 1
[5,2,2,2,1,1]
=> [6,4,1,1,1]
=> [4,3,2,2,1,1]
=> ? = 4 + 1
[5,2,2,1,1,1,1]
=> [7,4,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 3 + 1
[5,2,1,1,1,1,1,1]
=> [8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 2 + 1
[4,4,4,1]
=> [4,3,3,3]
=> [7,1,1,1,1,1,1]
=> ? = 3 + 1
[4,4,3,2]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 4 + 1
[4,4,2,1,1,1]
=> [6,3,3,1]
=> [8,2,1,1,1]
=> ? = 3 + 1
[4,4,1,1,1,1,1]
=> [7,3,3]
=> [8,1,1,1,1,1]
=> ? = 2 + 1
[4,3,3,1,1,1]
=> [6,3,2,2]
=> [3,3,3,2,1,1]
=> ? = 3 + 1
[4,3,2,2,1,1]
=> [6,3,2,1,1]
=> [7,3,1,1,1]
=> ? = 4 + 1
[4,3,2,1,1,1,1]
=> [7,3,2,1]
=> [6,2,1,1,1,1,1]
=> ? = 3 + 1
[4,2,2,2,2,1]
=> [6,3,1,1,1,1]
=> [4,3,3,1,1,1]
=> ? = 5 + 1
[4,2,2,2,1,1,1]
=> [7,3,1,1,1]
=> [4,4,1,1,1,1,1]
=> ? = 4 + 1
[4,2,1,1,1,1,1,1,1]
=> [9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> ? = 2 + 1
[4,1,1,1,1,1,1,1,1,1]
=> [10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,3,3,3,1]
=> [5,2,2,2,2]
=> [8,2,2,1]
=> ? = 4 + 1
[3,3,3,2,2]
=> [5,2,2,2,1,1]
=> [7,4,1,1]
=> ? = 5 + 1
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].
The following 128 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000288The number of ones in a binary word. St000251The number of nonsingleton blocks of a set partition. St000157The number of descents of a standard tableau. St000733The row containing the largest entry of a standard tableau. St000052The number of valleys of a Dyck path not on the x-axis. 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. St000291The number of descents of a binary word. St000146The Andrews-Garvan crank of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000292The number of ascents of a binary word. St000053The number of valleys of the Dyck path. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000996The number of exclusive left-to-right maxima of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000028The number of stack-sorts needed to sort a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000507The number of ascents of a standard tableau. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000204The number of internal nodes of a binary tree. St000065The number of entries equal to -1 in an alternating sign matrix. St000806The semiperimeter of the associated bargraph. St000711The number of big exceedences of a permutation. St000035The number of left outer peaks of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000245The number of ascents of a permutation. St000742The number of big ascents of a permutation after prepending zero. St001712The number of natural descents of a standard Young tableau. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St001462The number of factors of a standard tableaux under concatenation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000155The number of exceedances (also excedences) of 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. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001394The genus of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000647The number of big descents of a permutation. St000201The number of leaf nodes in a binary tree. St000672The number of minimal elements in Bruhat order not less than the permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000670The reversal length of a permutation. St000730The maximal arc length of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000098The chromatic number of a graph. St000232The number of crossings of a set partition. St000568The hook number of a binary tree. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000563The number of overlapping pairs of blocks of a set partition. St000884The number of isolated descents of a permutation. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001557The number of inversions of the second entry of a permutation. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001689The number of celebrities in a graph. St001746The coalition number of a graph. St001792The arboricity of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000779The tier of a permutation. St000353The number of inner valleys of a permutation. St001638The book thickness of a graph. St001737The number of descents of type 2 in a permutation. St000007The number of saliances of the permutation. St001644The dimension of a graph. St000354The number of recoils of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St001330The hat guessing number of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000710The number of big deficiencies of a permutation. St000872The number of very big descents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000021The number of descents of a permutation. St000092The number of outer peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000822The Hadwiger number of the graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000317The cycle descent number of a permutation. St000325The width of the tree associated to a permutation. St000486The number of cycles of length at least 3 of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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