Your data matches 255 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000849
(load all 2 compositions to match this statistic)
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000849: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 0
[1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 0
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
Description
The number of 1/3-balanced pairs in a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains. Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a $1/3$-balanced pair.
Matching statistic: St001388
(load all 14 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001388: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,2,3] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [1,2,3] => 0
[1,1,0,1,0,0]
 => [2,3,1] => [1,2,3] => 0
[1,1,1,0,0,0]
 => [3,2,1] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,2,3,4] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,3,2,4] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,3,4,2] => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [1,3,2,4] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,4,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,2,3,5,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [1,2,3,4,5,6] => 0
Description
The number of non-attacking neighbors of a permutation. For a permutation $\sigma$, the indices $i$ and $i+1$ are attacking if $|\sigma(i)-\sigma(i+1)| = 1$. Visually, this is, for $\sigma$ considered as a placement of kings on a chessboard, if the kings placed in columns $i$ and $i+1$ are non-attacking.
Matching statistic: St001397
(load all 2 compositions to match this statistic)
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 0
[1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 0
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
Description
Number of pairs of incomparable elements in a finite poset. For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
Matching statistic: St000100
(load all 2 compositions to match this statistic)
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
Description
The number of linear extensions of a poset.
Matching statistic: St001268
(load all 2 compositions to match this statistic)
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001268: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
Description
The size of the largest ordinal summand in the poset. The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$. This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.
Matching statistic: St001779
(load all 2 compositions to match this statistic)
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001779: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
Description
The order of promotion on the set of linear extensions of a poset.
Matching statistic: St000081
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
Description
The number of edges of a graph.
Matching statistic: St000171
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
Description
The degree of the graph. This is the maximal vertex degree of a graph.
Matching statistic: St000218
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [2,1] => [2,1] => 0
[1,1,0,0]
 => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [2,3,1] => [3,2,1] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [3,2,1] => [2,3,1] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0]
 => [3,2,1] => [2,1,3] => [2,1,3] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,4,3,1] => [3,4,2,1] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,2,4,1] => [2,4,3,1] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [4,2,3,1] => [2,3,4,1] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,2,1] => [3,2,4,1] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,1,4] => [2,3,1,4] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,5,4,1] => [4,5,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,4,3,5,1] => [3,5,4,2,1] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,5,3,4,1] => [3,4,5,2,1] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,5,4,3,1] => [4,3,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,2,4,5,1] => [2,5,4,3,1] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,2,5,4,1] => [2,4,5,3,1] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [4,2,3,5,1] => [2,3,5,4,1] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,2,4,3,1] => [2,4,3,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,3,4,5,2] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,5,4,2] => [1,4,5,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,4,3,5,2] => [1,3,5,4,2] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,3,4,6,5,1] => [5,6,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [2,3,5,4,6,1] => [4,6,5,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [2,3,6,4,5,1] => [4,5,6,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [2,4,3,5,6,1] => [3,6,5,4,2,1] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,4,3,6,5,1] => [3,5,6,4,2,1] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [2,5,3,4,6,1] => [3,4,6,5,2,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [2,6,3,4,5,1] => [3,4,5,6,2,1] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [3,2,4,5,6,1] => [2,6,5,4,3,1] => 0
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000271
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
Description
The chromatic index of a graph. This is the minimal number of colours needed such that no two adjacent edges have the same colour.
The following 245 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000289The decimal representation of a binary word. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000691The number of changes of a binary word. St000868The aid statistic in the sense of Shareshian-Wachs. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001341The number of edges in the center of a graph. St001402The number of separators in a permutation. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001649The length of a longest trail in a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000086The number of subgraphs. St000240The number of indices that are not small excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000388The number of orbits of vertices of a graph under automorphisms. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000983The length of the longest alternating subword. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001674The number of vertices of the largest induced star graph in the graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001725The harmonious chromatic number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000045The number of linear extensions of a binary tree. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000260The radius of a connected graph. St001890The maximum magnitude of the Möbius function of a poset. St000264The girth of a graph, which is not a tree. St001569The maximal modular displacement of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001095The number of non-isomorphic posets with precisely one further covering relation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000647The number of big descents of a permutation. St000731The number of double exceedences of a permutation. St001394The genus of a permutation. St000007The number of saliances of the permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St000635The number of strictly order preserving maps of a poset into itself. St001490The number of connected components of a skew partition. St000842The breadth of a permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000441The number of successions of a permutation. St000534The number of 2-rises of a permutation. St001118The acyclic chromatic index of a graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000451The length of the longest pattern of the form k 1 2. St000906The length of the shortest maximal chain in a poset. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001153The number of blocks with even minimum in a set partition. St000028The number of stack-sorts needed to sort a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001130The number of two successive successions in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001722The number of minimal chains with small intervals between a binary word and the top element. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000090The variation of a composition. St000091The descent variation of a composition. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000241The number of cyclical small excedances. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000370The genus of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000779The tier of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000962The 3-shifted major index of a permutation. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. St001411The number of patterns 321 or 3412 in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001839The number of excedances of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001871The number of triconnected components of a graph. St000352The Elizalde-Pak rank of a permutation. St000492The rob statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000989The number of final rises of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001344The neighbouring number of a permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001481The minimal height of a peak of a Dyck path. St001518The number of graphs with the same ordinary spectrum as the given graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001949The rigidity index of a graph. St000155The number of exceedances (also excedences) of a permutation. St000258The burning number of a graph. St000469The distinguishing number of a graph. St000482The (zero)-forcing number of a graph. St000542The number of left-to-right-minima of a permutation. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000839The largest opener of a set partition. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001366The maximal multiplicity of a degree of a vertex of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001623The number of doubly irreducible elements of a lattice. St001723The differential of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000230Sum of the minimal elements of the blocks of a set partition. St000686The finitistic dominant dimension of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001566The length of the longest arithmetic progression in a permutation. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.