Your data matches 153 different statistics following compositions of up to 3 maps.
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Mp00014: Binary trees to 132-avoiding permutationPermutations
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
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[[[.,.],.],[[.,.],.]]
=> [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.
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
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
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [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.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00064: Permutations reversePermutations
St000007: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [1,2] => 1
[[.,.],.]
=> [1,2] => [2,1] => 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 2
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [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].
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck 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
[.,[.,.]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [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.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 1
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
Description
The number of peaks of a Dyck path.
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
Mp00028: Dyck paths reverseDyck paths
St000025: 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]
=> [1,0,1,0]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 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]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [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 $D$.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
St000031: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [2,1] => 1
[[.,.],.]
=> [1,2] => [1,2] => 2
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 2
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [2,3,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,4,3,1] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,4,2,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,4,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,3,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [2,3,4,5,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,3,5,4,1] => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,4,5,3,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,5,4,3,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [2,5,3,4,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,4,2,5,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,5,2,4,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,3,5,2,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [4,5,2,3,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,4,2,1] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,4,2,3,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [3,1,4,5,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,1,5,4,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,1,5,3,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,1,4,3,2] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,1,3,4,2] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [2,4,1,5,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,5,1,4,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,5,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,1,2,4,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [2,3,5,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [3,5,2,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,5,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,2,3,4] => 1
Description
The number of cycles in the cycle decomposition of a permutation.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00069: Permutations complementPermutations
St000054: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [1,2] => 1
[[.,.],.]
=> [1,2] => [2,1] => 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => 2
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,3,1] => 2
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,3,4] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,1,4] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,1,4] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,4,2] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,3,2] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,4,1] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,4,1] => 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,4,3,1] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,4,2,1] => 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,3,4,5] => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,3,2,4,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,3,1,4,5] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,1,4,5] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,4,3,5] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,1,4,3,5] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,3,4,2,5] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,3,4,1,5] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,2,4,1,5] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,1,5] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,4,2,1,5] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,3,2,1,5] => 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,3,1,5,4] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,2,1,5,4] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,2,4,5,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,3,4,5,2] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,3,5,2] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,4,5,3,2] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
Mp00242: Dyck paths Hessenberg posetPosets
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
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[[[.,.],.],[[.,.],.]]
=> [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.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00069: Permutations complementPermutations
St000542: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [1,2] => 1
[[.,.],.]
=> [1,2] => [2,1] => 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => 2
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,3,1] => 2
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,3,4] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,1,4] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,1,4] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,4,2] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,3,2] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,4,1] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,4,1] => 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,4,3,1] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,4,2,1] => 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,3,4,5] => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,3,2,4,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,3,1,4,5] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,1,4,5] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,4,3,5] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,1,4,3,5] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,3,4,2,5] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,3,4,1,5] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,2,4,1,5] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,1,5] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,4,2,1,5] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,3,2,1,5] => 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,3,1,5,4] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,2,1,5,4] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,2,4,5,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,3,4,5,2] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,3,5,2] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,4,5,3,2] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 1
Description
The number of left-to-right-minima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
The following 143 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000678The number of up steps after the last double rise of a Dyck path. St000838The number of terminal right-hand endpoints when the vertices are written in order. 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. St001461The number of topologically connected components of the chord diagram of a permutation. St000053The number of valleys of the Dyck path. St000331The number of upper interactions of a Dyck path. St000439The position of the first down step of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 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. St000010The length of the partition. St000013The height of a Dyck path. St000026The position of the first return of a Dyck path. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000291The number of descents of a binary word. St000297The number of leading ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000382The first part of an integer composition. St000383The last part of an integer composition. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000617The number of global maxima of a Dyck path. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000759The smallest missing part in an integer partition. St000843The decomposition number of a perfect matching. St000883The number of longest increasing subsequences of a permutation. St000971The smallest closer of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. 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: St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) 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. St000245The number of ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000292The number of ascents of a binary word. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000445The number of rises of length 1 of a Dyck path. St000446The disorder of a permutation. St000504The cardinality of the first block of a set partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000546The number of global descents of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000738The first entry in the last row of a standard tableau. St000783The side length of the largest staircase partition fitting into a partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001484The number of singletons of an integer partition. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001777The number of weak descents in an integer composition. St000654The first descent 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. St000989The number of final rises of a permutation. St000061The number of nodes on the left branch of a binary tree. St000675The number of centered multitunnels of a Dyck path. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000653The last descent of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St001480The number of simple summands of the module J^2/J^3. St000159The number of distinct parts of the integer partition. St000993The multiplicity of the largest part of an integer partition. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001432The order dimension of the partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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. St000702The number of weak deficiencies of a permutation. 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. St001712The number of natural descents of a standard Young tableau.