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Your data matches 244 different statistics following compositions of up to 3 maps.
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Matching statistic: St001280
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 2
[3,1,1]
=> 1
[2,2,1]
=> 2
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 2
[3,2,1]
=> 2
[3,1,1,1]
=> 1
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 1
[4,3]
=> 2
[4,2,1]
=> 2
[4,1,1,1]
=> 1
[3,3,1]
=> 2
[3,2,2]
=> 3
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 1
[5,3]
=> 2
[5,2,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000147
(load all 4 compositions to match this statistic)
(load all 4 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)
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1 = 0 + 1
[2]
=> [1,1]
=> 2 = 1 + 1
[1,1]
=> [2]
=> 1 = 0 + 1
[3]
=> [2,1]
=> 2 = 1 + 1
[2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,1]
=> [3]
=> 1 = 0 + 1
[4]
=> [3,1]
=> 2 = 1 + 1
[3,1]
=> [2,2]
=> 2 = 1 + 1
[2,2]
=> [2,1,1]
=> 3 = 2 + 1
[2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,1,1,1]
=> [4]
=> 1 = 0 + 1
[5]
=> [4,1]
=> 2 = 1 + 1
[4,1]
=> [3,2]
=> 2 = 1 + 1
[3,2]
=> [2,2,1]
=> 3 = 2 + 1
[3,1,1]
=> [3,2]
=> 2 = 1 + 1
[2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
[2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [5]
=> 1 = 0 + 1
[6]
=> [5,1]
=> 2 = 1 + 1
[5,1]
=> [4,2]
=> 2 = 1 + 1
[4,2]
=> [3,2,1]
=> 3 = 2 + 1
[4,1,1]
=> [3,3]
=> 2 = 1 + 1
[3,3]
=> [2,2,2]
=> 3 = 2 + 1
[3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[3,1,1,1]
=> [4,2]
=> 2 = 1 + 1
[2,2,2]
=> [3,1,1,1]
=> 4 = 3 + 1
[2,2,1,1]
=> [4,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [5,1]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [6]
=> 1 = 0 + 1
[7]
=> [6,1]
=> 2 = 1 + 1
[6,1]
=> [5,2]
=> 2 = 1 + 1
[5,2]
=> [4,2,1]
=> 3 = 2 + 1
[5,1,1]
=> [4,3]
=> 2 = 1 + 1
[4,3]
=> [3,2,2]
=> 3 = 2 + 1
[4,2,1]
=> [3,3,1]
=> 3 = 2 + 1
[4,1,1,1]
=> [4,3]
=> 2 = 1 + 1
[3,3,1]
=> [3,2,2]
=> 3 = 2 + 1
[3,2,2]
=> [3,2,1,1]
=> 4 = 3 + 1
[3,2,1,1]
=> [4,2,1]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [5,2]
=> 2 = 1 + 1
[2,2,2,1]
=> [4,1,1,1]
=> 4 = 3 + 1
[2,2,1,1,1]
=> [5,1,1]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [6,1]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [7]
=> 1 = 0 + 1
[8]
=> [7,1]
=> 2 = 1 + 1
[7,1]
=> [6,2]
=> 2 = 1 + 1
[6,2]
=> [5,2,1]
=> 3 = 2 + 1
[6,1,1]
=> [5,3]
=> 2 = 1 + 1
[5,3]
=> [4,2,2]
=> 3 = 2 + 1
[5,2,1]
=> [4,3,1]
=> 3 = 2 + 1
Description
The length of the partition.
Matching statistic: St000378
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
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}
[2,2,2,2,1,1,1,1,1,1,1]
=> [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}
[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]
=> ? ∊ {2,2,2,2}
[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]
=> ? ∊ {2,2,2,2}
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> [10,1,1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[3,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[3,3,3,3,1,1,1,1]
=> [3,3,3,1,1,1,1]
=> [7,3,3]
=> [8,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[3,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> [11,1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[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]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
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: 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: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
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]
=> ? ∊ {1,1,2,2,3,3,3,3}
[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]
=> ? ∊ {1,1,2,2,3,3,3,3}
[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]
=> ? ∊ {1,1,2,2,3,3,3,3}
[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]
=> ? ∊ {1,1,2,2,3,3,3,3}
[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]
=> ? ∊ {1,1,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {1,1,2,2,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ?
=> ? ∊ {1,1,2,2,3,3,3,3}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[3,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[5,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[5,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[5,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
[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]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5}
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: St000394
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 89%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 89%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 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]
=> [1,1,0,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 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]
=> [1,0,1,0,1,1,0,0]
=> 1
[5]
=> []
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 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]
=> [1,1,1,0,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[6]
=> []
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 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]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[7]
=> []
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 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]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[8]
=> []
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 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]
=> [1,0,1,1,0,1,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1}
[13]
=> []
=> []
=> []
=> ? ∊ {0,1,1,2,2,2}
[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]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,2,2,2}
[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]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,2,2,2}
[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]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,2,2,2}
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,2,2,2}
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {0,1,1,2,2,2}
[14]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[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]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[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]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[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]
=> ?
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[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]
=> ?
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3,3,4}
[15]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[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]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[3,3,2,2,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[3,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[3,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[3,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[3,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000288
(load all 7 compositions to match this statistic)
(load all 7 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: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 89%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 89%
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]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> 100000000000 => ? ∊ {0,1}
[13]
=> []
=> []
=> => ? ∊ {0,1,1,2,2}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> 1000001000 => ? ∊ {0,1,1,2,2}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? ∊ {0,1,1,2,2}
[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 => ? ∊ {0,1,1,2,2}
[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]
=> [12]
=> 1000000000000 => ? ∊ {0,1,1,2,2}
[14]
=> []
=> []
=> => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> 1000001100 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> 1000010010 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> 100000000110 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> 1000001000 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[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 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> 1000010000 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [9,3]
=> 10000001000 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [10,2]
=> 100000000100 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[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 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> 1000000000000 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> 10000000000000 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3,3,3}
[15]
=> []
=> []
=> => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [6,2,2,1]
=> 1000011010 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [6,3,1,1]
=> 1000100110 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> 100000001110 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> 1000001100 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> 1000010010 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> 100000000110 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> 1000001000 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[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 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[3,3,3,3,1,1,1]
=> [3,3,3,1,1,1]
=> [6,3,3]
=> 100011000 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [7,3,2]
=> 1000010100 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> 10000001100 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[3,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [6,5,1]
=> 101000010 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [7,4,1]
=> 1000100010 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[3,3,2,2,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1]
=> [8,3,1]
=> 10000010010 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [9,2,1]
=> 100000001010 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[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 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
[3,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> 1000010000 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
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: 78% ●values known / values provided: 85%●distinct values known / distinct values provided: 78%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 85%●distinct values known / distinct values provided: 78%
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]
=> [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]
=> ? ∊ {0,1}
[10]
=> []
=> []
=> []
=> ? ∊ {0,1,1}
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1}
[11]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,2}
[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]
=> ? ∊ {0,1,1,1,2}
[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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2}
[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]
=> ? ∊ {0,1,1,1,2}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2}
[12]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
[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]
=> ? ∊ {0,1,1,1,1,2,2,2}
[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]
=> ? ∊ {0,1,1,1,1,2,2,2}
[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]
=> ? ∊ {0,1,1,1,1,2,2,2}
[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]
=> ? ∊ {0,1,1,1,1,2,2,2}
[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]
=> ? ∊ {0,1,1,1,1,2,2,2}
[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]
=> ? ∊ {0,1,1,1,1,2,2,2}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {0,1,1,1,1,2,2,2}
[13]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3}
[14]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[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]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
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: St000734
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 89%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 89%
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]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? ∊ {0,1}
[13]
=> []
=> []
=> ? ∊ {0,1,1,2,2,2,2}
[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]]
=> ? ∊ {0,1,1,2,2,2,2}
[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]]
=> ? ∊ {0,1,1,2,2,2,2}
[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]]
=> ? ∊ {0,1,1,2,2,2,2}
[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]]
=> ? ∊ {0,1,1,2,2,2,2}
[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]]
=> ? ∊ {0,1,1,2,2,2,2}
[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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? ∊ {0,1,1,2,2,2,2}
[14]
=> []
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[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]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[15]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[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]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[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]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[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]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[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]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[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]]
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000733
(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
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 89%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 89%
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]
=> [[1,2]]
=> 1
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[2,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[5]
=> []
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[3,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[6]
=> []
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[4,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[3,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[2,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[7]
=> []
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[5,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[4,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[3,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 1
[8]
=> []
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[6,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[5,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[4,4]
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[4,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[3,3,1,1]
=> [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]
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? ∊ {0,1}
[13]
=> []
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2}
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? ∊ {0,1,2,2,2,2,2}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? ∊ {0,1,2,2,2,2,2}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? ∊ {0,1,2,2,2,2,2}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? ∊ {0,1,2,2,2,2,2}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? ∊ {0,1,2,2,2,2,2}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? ∊ {0,1,2,2,2,2,2}
[14]
=> []
=> []
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [11,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[15]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
Description
The row containing the largest entry of a standard tableau.
The following 234 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000141The maximum drop size of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000382The first part of an integer composition. St000691The number of changes of a binary word. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000381The largest part of an integer composition. 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. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000808The number of up steps of the associated bargraph. St000653The last descent of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001480The number of simple summands of the module J^2/J^3. 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. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. 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. St000840The number of closers smaller than the largest opener in a perfect matching. St000740The last entry of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000308The height of the tree associated to a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001667The maximal size of a pair of weak twins for a permutation. 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. St001517The length of a longest pair of twins in a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001060The distinguishing index of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000264The girth of a graph, which is not a tree. St001487The number of inner corners of a skew partition. St000157The number of descents of a standard tableau. St000259The diameter of a connected graph. St000624The normalized sum of the minimal distances to a greater element. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000831The number of indices that are either descents or recoils. St000837The number of ascents of distance 2 of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000662The staircase size of the code of a permutation. St000354The number of recoils of a permutation. St000306The bounce count of a 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. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000352The Elizalde-Pak rank of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. 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. St000374The number of exclusive right-to-left minima of a permutation. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St000451The length of the longest pattern of the form k 1 2. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000507The number of ascents of a standard tableau. St000651The maximal size of a rise in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St001928The number of non-overlapping descents in a permutation. St000542The number of left-to-right-minima of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000242The number of indices that are not cyclical small weak excedances. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000710The number of big deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000325The width of the tree associated to a permutation. St000619The number of cyclic descents of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000638The number of up-down runs of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000052The number of valleys of a Dyck path not on the x-axis. St000260The radius of a connected graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000035The number of left outer peaks of a permutation. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000331The number of upper interactions of a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001394The genus of a permutation. St001729The number of visible descents of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000365The number of double ascents of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001874Lusztig's a-function for the symmetric group. St000166The depth minus 1 of an ordered tree. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000982The length of the longest constant subword. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000094The depth of an ordered tree. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001717The largest size of an interval in a poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000670The reversal length of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St000730The maximal arc length of a set partition. St000884The number of isolated descents of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000007The number of saliances of the permutation. St001557The number of inversions of the second entry of a permutation. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St001470The cyclic holeyness of a permutation. St000092The number of outer peaks of a permutation. St000120The number of left tunnels of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001955The number of natural descents for set-valued two row standard Young tableaux. 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 $eA$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001298The number of repeated entries in the Lehmer code of a permutation. St001589The nesting number of a perfect matching. St000989The number of final rises of a permutation. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001469The holeyness of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000080The rank of the poset. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000332The positive inversions of an alternating sign matrix. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001716The 1-improper chromatic number of a graph. St001792The arboricity of a graph. St000024The number of double up and double down steps of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001948The number of augmented double ascents of a permutation. St001569The maximal modular displacement of a permutation. St001668The number of points of the poset minus the width of the poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000628The balance of a binary word. St000456The monochromatic index of a connected graph. St001488The number of corners of a skew partition. St001896The number of right descents of a signed permutations. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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