Your data matches 94 different statistics following compositions of up to 3 maps.
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Mp00051: Ordered trees to Dyck pathDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Mp00050: Ordered trees to binary tree: right brother = right childBinary trees
Mp00014: Binary trees to 132-avoiding permutationPermutations
St000007: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => 2
[[[]]]
=> [[.,.],.]
=> [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => 2
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 4
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 4
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 3
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 3
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 3
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 2
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 4
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 3
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 3
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 3
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 2
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
St000843: Perfect matchings ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [(1,2)]
=> 1
[[],[]]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[[]]]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2
Description
The decomposition number of a perfect matching. This is the number of integers $i$ such that all elements in $\{1,\dots,i\}$ are matched among themselves. Visually, it is the number of components of the arc diagram of the matching, where a component is a matching of a set of consecutive numbers $\{a,a+1,\dots,b\}$ such that there is no arc matching a number smaller than $a$ with a number larger than $b$. E.g., $\{(1,6),(2,4),(3,5)\}$ is a hairpin under a single edge - crossing nested by a single arc. Thus, this matching has one component. However, $\{(1,2),(3,6),(4,5)\}$ is a single edge to the left of a ladder (a pair of nested edges), so it has two components.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St000237: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => 0 = 1 - 1
[[],[]]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 2 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0 = 1 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3 = 4 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 2 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 2 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0 = 1 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0 = 1 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0 = 1 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4 = 5 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 3 = 4 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3 = 4 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 2 = 3 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2 = 3 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3 = 4 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2 = 3 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 2 = 3 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2 = 3 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1 = 2 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3 = 4 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 2 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 2 = 3 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1 = 2 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1 = 2 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1 = 2 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1 = 2 - 1
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
Description
The position of the first down step of a Dyck path.
Mp00050: Ordered trees to binary tree: right brother = right childBinary trees
Mp00014: Binary trees to 132-avoiding permutationPermutations
St000546: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => 0 = 1 - 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[[[]]]
=> [[.,.],.]
=> [1,2] => 0 = 1 - 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 2 = 3 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1 = 2 - 1
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1 = 2 - 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0 = 1 - 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 4 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2 = 3 - 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2 = 3 - 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1 = 2 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1 = 2 - 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 3 - 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1 = 2 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 2 - 1
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 2 - 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0 = 1 - 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0 = 1 - 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0 = 1 - 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0 = 1 - 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 4 = 5 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 3 = 4 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 3 = 4 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2 = 3 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2 = 3 - 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 3 = 4 - 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2 = 3 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2 = 3 - 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 2 = 3 - 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1 = 2 - 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1 = 2 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1 = 2 - 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1 = 2 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1 = 2 - 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 3 = 4 - 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2 = 3 - 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2 = 3 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1 = 2 - 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1 = 2 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2 = 3 - 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2 = 3 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1 = 2 - 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1 = 2 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1 = 2 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1 = 2 - 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1 = 2 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1 = 2 - 1
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1 = 2 - 1
Description
The number of global descents of a permutation. The global descents are the integers in the set $$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$ In particular, if $i\in C(\pi)$ then $i$ is a descent. For the number of global ascents, see [[St000234]].
Matching statistic: St000010
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => [1]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2
[[[]]]
=> [1,1,0,0]
=> [2] => [2]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
Description
The length of the partition.
Matching statistic: St000097
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000097: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => ([],1)
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[[[]]]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
Description
The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
Mp00094: Integer compositions to binary wordBinary words
St000288: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => 1 => 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => 11 => 2
[[[]]]
=> [1,1,0,0]
=> [2] => 10 => 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3] => 100 => 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 10001 => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 10001 => 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 2
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Mp00049: Ordered trees to binary tree: left brother = left childBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1,0]
=> [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,1,0,0]
=> [2] => 2
[[[]]]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 3
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 4
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 3
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 2
Description
The first part of an integer composition.
The following 84 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001581The achromatic number of a graph. St000383The last part of an integer composition. St000925The number of topologically connected components of a set partition. St001050The number of terminal closers of a set partition. St000098The chromatic number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000068The number of minimal elements in a poset. St000025The number of initial rises of a Dyck path. St000234The number of global ascents of a permutation. St000172The Grundy number of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St000971The smallest closer of a set partition. St001029The size of the core of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000504The cardinality of the first block of a set partition. St000502The number of successions of a set partitions. St000069The number of maximal elements of a poset. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000054The first entry of the permutation. St000908The length of the shortest maximal antichain in a poset. St000654The first descent of a permutation. St000914The sum of the values of the Möbius function of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St000989The number of final rises of a permutation. St000740The last entry of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000084The number of subtrees. St000056The decomposition (or block) number of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000822The Hadwiger number of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000133The "bounce" of a permutation. St000203The number of external nodes of a binary tree. St000738The first entry in the last row of a standard tableau. St000883The number of longest increasing subsequences of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000648The number of 2-excedences of a permutation. St000924The number of topologically connected components of a perfect matching. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001621The number of atoms of a lattice. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000898The number of maximal entries in the last diagonal of the monotone triangle. St001889The size of the connectivity set of a signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.