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Your data matches 71 different statistics following compositions of up to 3 maps.
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Matching statistic: St001184
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001184: 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]
=> 1
[[[]]]
=> [1,1,0,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[[[],[]],[[]]]
=> [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]
=> 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St000007
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,2] => 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => 2
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 3
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 2
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 3
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 4
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 3
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
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].
Matching statistic: St000011
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
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.
Matching statistic: St000025
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000025: 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]
=> 1
[[[]]]
=> [[[]]]
=> [1,1,0,0]
=> 2
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> 2
[[[]],[]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> 1
[[[],[]]]
=> [[[],[]]]
=> [1,1,0,1,0,0]
=> 2
[[[[]]]]
=> [[[[]]]]
=> [1,1,1,0,0,0]
=> 3
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[],[[]],[]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 3
[[[]],[],[]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[]],[[]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[[[]]],[]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[],[],[]]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[],[[]]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[[],[]]]]
=> [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[[[[]]]]]
=> [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000068
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> ([],1)
=> 1
[[],[]]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[[[]]]
=> [1,1,0,0]
=> ([],2)
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> ([],3)
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
Description
The number of minimal elements in a poset.
Matching statistic: St000084
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [[]]
=> 1
[[],[]]
=> [[],[]]
=> [[[]]]
=> 1
[[[]]]
=> [[[]]]
=> [[],[]]
=> 2
[[],[],[]]
=> [[],[],[]]
=> [[[[]]]]
=> 1
[[],[[]]]
=> [[[]],[]]
=> [[],[[]]]
=> 2
[[[]],[]]
=> [[],[[]]]
=> [[[],[]]]
=> 1
[[[],[]]]
=> [[[],[]]]
=> [[[]],[]]
=> 2
[[[[]]]]
=> [[[[]]]]
=> [[],[],[]]
=> 3
[[],[],[],[]]
=> [[],[],[],[]]
=> [[[[[]]]]]
=> 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [[],[[[]]]]
=> 2
[[],[[]],[]]
=> [[],[[]],[]]
=> [[[],[[]]]]
=> 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [[[]],[[]]]
=> 2
[[],[[[]]]]
=> [[[[]]],[]]
=> [[],[],[[]]]
=> 3
[[[]],[],[]]
=> [[],[],[[]]]
=> [[[[],[]]]]
=> 1
[[[]],[[]]]
=> [[[]],[[]]]
=> [[],[[],[]]]
=> 2
[[[],[]],[]]
=> [[],[[],[]]]
=> [[[[]],[]]]
=> 1
[[[[]]],[]]
=> [[],[[[]]]]
=> [[[],[],[]]]
=> 1
[[[],[],[]]]
=> [[[],[],[]]]
=> [[[[]]],[]]
=> 2
[[[],[[]]]]
=> [[[[]],[]]]
=> [[],[[]],[]]
=> 3
[[[[]],[]]]
=> [[[],[[]]]]
=> [[[],[]],[]]
=> 2
[[[[],[]]]]
=> [[[[],[]]]]
=> [[[]],[],[]]
=> 3
[[[[[]]]]]
=> [[[[[]]]]]
=> [[],[],[],[]]
=> 4
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [[[[[[]]]]]]
=> 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [[],[[[[]]]]]
=> 2
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [[[],[[[]]]]]
=> 1
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [[[]],[[[]]]]
=> 2
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [[],[],[[[]]]]
=> 3
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [[[[],[[]]]]]
=> 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [[],[[],[[]]]]
=> 2
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [[[[]],[[]]]]
=> 1
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [[[],[],[[]]]]
=> 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [[[[]]],[[]]]
=> 2
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [[],[[]],[[]]]
=> 3
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [[[],[]],[[]]]
=> 2
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [[[]],[],[[]]]
=> 3
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [[],[],[],[[]]]
=> 4
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [[[[[],[]]]]]
=> 1
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [[],[[[],[]]]]
=> 2
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [[[],[[],[]]]]
=> 1
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [[[]],[[],[]]]
=> 2
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [[],[],[[],[]]]
=> 3
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [[[[[]],[]]]]
=> 1
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [[[[],[],[]]]]
=> 1
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [[],[[[]],[]]]
=> 2
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [[],[[],[],[]]]
=> 2
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [[[[[]]],[]]]
=> 1
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [[[],[[]],[]]]
=> 1
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [[[[],[]],[]]]
=> 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [[[[]],[],[]]]
=> 1
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [[[],[],[],[]]]
=> 1
Description
The number of subtrees.
Matching statistic: St000314
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => 1
[[[]]]
=> [[.,.],.]
=> [1,2] => 2
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 3
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 3
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 2
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 3
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 2
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 3
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 4
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 3
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000838
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000838: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000838: 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)]
=> 1
[[[]]]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[[],[[]]]
=> [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)]
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
[[],[[],[]]]
=> [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)]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[[[]],[[]]]
=> [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)]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> 2
[[],[],[[[]]]]
=> [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)]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 1
[[],[[],[],[]]]
=> [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)]
=> 3
[[],[[[]],[]]]
=> [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)]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[[[]],[[],[]]]
=> [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)]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 1
[[[],[]],[[]]]
=> [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)]
=> 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 1
Description
The number of terminal right-hand endpoints when the vertices are written in order.
An opener (or left hand endpoint) of a perfect matching is a number that is matched with a larger number, which is then called a closer (or right hand endpoint).
The opener-closer sequence of the perfect matching $\{(1,3),(2,5),(4,6)\}$ is $OOCOCC$, so the number of terminal right-hand endpoints is $2$.
The number of perfect matchings of $\{1,\dots,2n\}$ with exactly $T$ terminal closers, according to [1] computed in [2], is
$$
\frac{T(2n-T-1)!}{2^{n-T}(n-T)!}.
$$
Matching statistic: St000439
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [1,0]
=> 2 = 1 + 1
[[],[]]
=> [[],[]]
=> [1,0,1,0]
=> 2 = 1 + 1
[[[]]]
=> [[[]]]
=> [1,1,0,0]
=> 3 = 2 + 1
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[[]],[]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[[],[]]]
=> [[[],[]]]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[[[[]]]]
=> [[[[]]]]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[],[[]],[]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[]],[],[]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[[]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[[[]]],[]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[],[],[]]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[[],[[]]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[[[],[]]]]
=> [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[[[[]]]]]
=> [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000546
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => 0 = 1 - 1
[[],[]]
=> [[.,.],.]
=> [1,2] => 0 = 1 - 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1 = 2 - 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0 = 1 - 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1 = 2 - 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 2 = 3 - 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 2 - 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0 = 1 - 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1 = 2 - 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 3 - 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0 = 1 - 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 2 - 1
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0 = 1 - 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0 = 1 - 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1 = 2 - 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2 = 3 - 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1 = 2 - 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2 = 3 - 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 4 - 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0 = 1 - 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1 = 2 - 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0 = 1 - 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1 = 2 - 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2 = 3 - 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 0 = 1 - 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1 = 2 - 1
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 0 = 1 - 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 0 = 1 - 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1 = 2 - 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2 = 3 - 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1 = 2 - 1
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2 = 3 - 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 3 = 4 - 1
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0 = 1 - 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1 = 2 - 1
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0 = 1 - 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1 = 2 - 1
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2 = 3 - 1
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 0 = 1 - 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 0 = 1 - 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1 = 2 - 1
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1 = 2 - 1
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0 = 1 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 0 = 1 - 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0 = 1 - 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0 = 1 - 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0 = 1 - 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]].
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000286The number of connected components of the complement of a graph. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000740The last entry of a permutation. St000759The smallest missing part in an integer partition. St000843The decomposition number of a perfect matching. St000908The length of the shortest maximal antichain in a poset. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000053The number of valleys of the Dyck path. St000133The "bounce" of a permutation. St000203The number of external nodes of a binary tree. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000331The number of upper interactions of a Dyck path. St000504The cardinality of the first block of a set partition. St000738The first entry in the last row of a standard tableau. St001169Number of simple modules with projective dimension at least two 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. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000061The number of nodes on the left branch of a binary tree. St000654The first descent of a permutation. St000989The number of final rises of a permutation. St000675The number of centered multitunnels of a Dyck path. St000717The number of ordinal summands of a poset. St000914The sum of the values of the Möbius function of a poset. 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. St000993The multiplicity of the largest part of an integer partition. 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. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000942The number of critical left to right maxima of the parking functions. 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. St001904The length of the initial strictly increasing segment of a parking function.
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