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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00306: Posets —rowmotion cycle type⟶ 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)
=> [2]
=> 2
([],2)
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> 3
([],3)
=> [2,2,2,2]
=> 2
([(1,2)],3)
=> [6]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> 3
([(2,3)],4)
=> [6,6]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 4
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> 7
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 5
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 5
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> 5
Description
The largest part of an integer partition.
Matching statistic: St000474
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000474: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 2
([],2)
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> 3
([],3)
=> [2,2,2,2]
=> 2
([(1,2)],3)
=> [6]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> 3
([(2,3)],4)
=> [6,6]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 4
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> 7
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 5
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 5
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> 5
Description
Dyson's crank of a partition.
Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as
$$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> 2
([],2)
=> [2,2]
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 3
([],3)
=> [2,2,2,2]
=> [4,4]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 3
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 4
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 7
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 5
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 5
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 5
Description
The length of the partition.
Matching statistic: St000013
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 4
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> 7
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 5
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 10
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8
([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8
([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,5),(0,6),(1,5),(1,6),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(5,2),(6,3),(6,4)],7)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6
([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
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: St000643
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ? = 2
([],2)
=> 2
([(0,1)],2)
=> 3
([],3)
=> 2
([(1,2)],3)
=> 6
([(0,1),(0,2)],3)
=> 3
([(0,2),(2,1)],3)
=> 4
([(0,2),(1,2)],3)
=> 3
([(2,3)],4)
=> 6
([(1,2),(1,3)],4)
=> 6
([(0,1),(0,2),(0,3)],4)
=> 3
([(0,2),(0,3),(3,1)],4)
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,2),(2,3)],4)
=> 4
([(0,3),(3,1),(3,2)],4)
=> 4
([(1,3),(2,3)],4)
=> 6
([(0,3),(1,3),(3,2)],4)
=> 4
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 3
([(0,3),(1,2),(1,3)],4)
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,3),(2,1),(3,2)],4)
=> 5
([(0,3),(1,2),(2,3)],4)
=> 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(4,1),(4,2)],5)
=> 7
([(1,2),(1,3),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(1,4),(4,2),(4,3)],5)
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> 4
([(1,4),(2,4),(4,3)],5)
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> 5
([(0,4),(1,3),(2,3),(3,4)],5)
=> 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> 5
([(0,4),(1,2),(1,3),(3,4)],5)
=> 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 6
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
=> ? = 5
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 6
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> ? = 12
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 7
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 6
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 7
([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
=> ? = 9
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> ? = 6
([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
=> ? = 9
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ? = 6
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 12
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
=> ? = 9
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> ? = 12
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 6
([(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ? = 8
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> ? = 12
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 6
([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
=> ? = 9
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 6
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 6
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> ? = 12
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 6
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 6
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 10
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 6
([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> ? = 12
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
=> ? = 5
([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
=> ? = 6
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 12
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 10
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> ? = 12
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> ? = 12
([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
=> ? = 9
([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> ? = 12
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
=> ? = 9
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 6
([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 6
([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ? = 9
([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ? = 7
([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ? = 7
([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ? = 12
([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> ? = 12
([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> ? = 12
([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7)
=> ? = 6
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ? = 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 8
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 10
Description
The size of the largest orbit of antichains under Panyushev complementation.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 82%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 82%
Values
([],1)
=> [2]
=> [1,1]
=> 110 => 2
([],2)
=> [2,2]
=> [2,2]
=> 1100 => 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1110 => 3
([],3)
=> [2,2,2,2]
=> [4,4]
=> 110000 => 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 1111110 => 6
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 3
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 11110 => 4
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 3
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> 11111100 => 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 4
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> 111100 => 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> 111000 => 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> 110010 => 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 111110 => 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? = 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 4
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? = 7
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 1110010 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 11000010 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 5
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? = 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 110111110 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 5
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? = 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 11000010 => 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 10
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 8
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 110111110 => 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 10
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> 110000110 => ? = 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 8
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> 110000110 => ? = 4
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> 110000110 => ? = 4
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> 110000110 => ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> 110000110 => ? = 4
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 8
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 8
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 8
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> [8,4]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 8
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 9
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 9
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 8
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> 110000110 => ? = 4
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> 111110110 => ? = 7
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,4]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 8
([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 9
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 8
([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> 111110110 => ? = 7
([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 9
([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> 111110110 => ? = 7
([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> 111110110 => ? = 7
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [6,2,2,2]
=> [4,4,1,1,1,1]
=> 1100011110 => ? = 6
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [6,2,2,2]
=> [4,4,1,1,1,1]
=> 1100011110 => ? = 6
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,5),(0,6),(1,5),(1,6),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 8
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(5,2),(6,3),(6,4)],7)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 8
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [6,2,2,2]
=> [4,4,1,1,1,1]
=> 1100011110 => ? = 6
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 8
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000439
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 91%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 91%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 7 = 6 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 8 = 7 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 6 = 5 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 8 = 7 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 7 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 6 = 5 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 6 = 5 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5 = 4 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 5 = 4 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 5 = 4 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5 = 4 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 6 = 5 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 7 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> 9 = 8 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 6 = 5 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> 8 = 7 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5 = 4 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 11 = 10 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 6 = 5 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> 11 = 10 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 9 = 8 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 7 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 11 = 10 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> 9 = 8 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> 8 = 7 + 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 11 = 10 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 6 = 5 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 9 = 8 + 1
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 9 = 8 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8 + 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8 + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8 + 1
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8 + 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> ? = 8 + 1
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> ? = 9 + 1
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> ? = 9 + 1
([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> ? = 8 + 1
([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> ? = 9 + 1
([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8 + 1
([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> ? = 9 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 + 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 + 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,5),(0,6),(1,5),(1,6),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8 + 1
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(5,2),(6,3),(6,4)],7)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 8 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 + 1
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [6,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 6 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000734
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 91%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 91%
Values
([],1)
=> [2]
=> [[1,2]]
=> 2
([],2)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,1)],2)
=> [3]
=> [[1,2,3]]
=> 3
([],3)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 2
([(1,2)],3)
=> [6]
=> [[1,2,3,4,5,6]]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? = 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> 4
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? = 7
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 5
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? = 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> 5
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? = 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? = 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> 10
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 8
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 5
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 8
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 8
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 4
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 4
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 4
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 8
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 8
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 8
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> [8,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12]]
=> ? = 8
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? = 9
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? = 9
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 8
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 4
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12]]
=> ? = 8
([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? = 9
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 8
([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? = 9
([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
=> [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? = 5
([(0,4),(0,5),(1,6),(4,6),(5,1),(6,2),(6,3)],7)
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 9
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? = 6
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? = 6
([(0,6),(1,6),(4,3),(5,2),(5,4),(6,5)],7)
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 9
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
=> [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? = 5
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
=> [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? = 5
([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 9
([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 9
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
=> [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? = 5
([(0,5),(0,6),(1,5),(1,6),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 8
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(5,2),(6,3),(6,4)],7)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 8
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000676
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 64%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 64%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 8
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 8
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 8
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9
([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,4),(0,5),(1,6),(4,6),(5,1),(6,2),(6,3)],7)
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6
([(0,6),(1,6),(4,3),(5,2),(5,4),(6,5)],7)
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9
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)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 64%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 64%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 8
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 8
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 8
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9
([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,4),(0,5),(1,6),(4,6),(5,1),(6,2),(6,3)],7)
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6
([(0,6),(1,6),(4,3),(5,2),(5,4),(6,5)],7)
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. 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. St001809The index of the step at the first peak of maximal height in a Dyck path. 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. St000874The position of the last double rise in a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000738The first entry in the last row of a standard tableau. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one.
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