Your data matches 118 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000183
(load all 2 compositions to match this statistic)
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => 1 = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 2 = 1 + 1
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 2 = 1 + 1
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 2 = 1 + 1
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
Description
The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => 1 = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 2 = 1 + 1
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 2 = 1 + 1
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 2 = 1 + 1
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => 1 = 0 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000480
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1]
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1]
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1]
 => 0
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1]
 => 0
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,1]
 => 0
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [2]
 => 1
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [2]
 => 1
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [2]
 => 1
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [2]
 => 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
Description
The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000660
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
Description
The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000661
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
Description
The number of rises of length 3 of a Dyck path.
Matching statistic: St000931
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
Description
The number of occurrences of the pattern UUU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001251
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1]
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1]
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1]
 => 0
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1]
 => 0
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1]
 => 0
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,1]
 => 0
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [2]
 => 1
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [2]
 => 1
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [2]
 => 1
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [2]
 => 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 0
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St000442
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000920
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Matching statistic: St000013
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
[.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[.,[[[.,.],.],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
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 $i$.
The following 108 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000444The length of the maximal rise of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000758The length of the longest staircase fitting into an integer composition. St001597The Frobenius rank of a skew partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001964The interval resolution global dimension of a poset. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001110The 3-dynamic chromatic number of a graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St001271The competition number of a graph. St001330The hat guessing number of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001618The cardinality of the Frattini sublattice of a lattice. St001644The dimension of a graph. St001638The book thickness of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001734The lettericity of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001625The Möbius invariant of a lattice. St000095The number of triangles of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001871The number of triconnected components of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001828The Euler characteristic of a graph. St000822The Hadwiger number of the graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000379The number of Hamiltonian cycles in a graph. St001307The number of induced stars on four vertices in a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001367The smallest number which does not occur as degree of a vertex in a graph. St000286The number of connected components of the complement of a graph. St000456The monochromatic index of a connected graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001339The irredundance number of a graph. St001315The dissociation number of a graph.