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Your data matches 364 different statistics following compositions of up to 3 maps.
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Matching statistic: St000009
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(load all 3 compositions to match this statistic)
Mp00014: Binary trees āto 132-avoiding permutationā¶ Permutations
Mp00059: Permutations āRobinson-Schensted insertion tableauā¶ Standard tableaux
St000009: Standard tableaux ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00059: Permutations āRobinson-Schensted insertion tableauā¶ Standard tableaux
St000009: Standard tableaux ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [[1]]
=> 0
[.,[.,.]]
=> [2,1] => [[1],[2]]
=> 0
[[.,.],.]
=> [1,2] => [[1,2]]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [[1],[2],[3]]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [[1,3],[2]]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [[1,2],[3]]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 3
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[1,3],[2],[4]]
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[1,5],[2],[3],[4]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[1,4],[2],[3],[5]]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[1,5],[2],[3],[4]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[1,5],[2],[3],[4]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[1,3],[2],[4],[5]]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[1,4],[2],[3],[5]]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5]]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [[1,5],[2],[3],[4],[6]]
=> 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [[1,6],[2],[3],[4],[5]]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6]]
=> 3
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [[1,5],[2],[3],[4],[6]]
=> 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [[1,6],[2],[3],[4],[5]]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [6,5,4,2,3,1] => [[1,3],[2],[4],[5],[6]]
=> 4
[.,[[.,[.,.]],[.,[.,.]]]]
=> [6,5,3,2,4,1] => [[1,4],[2],[3],[5],[6]]
=> 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [6,4,3,2,5,1] => [[1,5],[2],[3],[4],[6]]
=> 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [5,4,3,2,6,1] => [[1,6],[2],[3],[4],[5]]
=> 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[[.,[.,.]],[.,[.,[.,.]]]]
=> [6,5,4,2,1,3] => [[1,3],[2],[4],[5],[6]]
=> 4
[[.,[.,[.,.]]],[.,[.,.]]]
=> [6,5,3,2,1,4] => [[1,4],[2],[3],[5],[6]]
=> 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [6,4,3,2,1,5] => [[1,5],[2],[3],[4],[6]]
=> 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [5,4,3,2,1,6] => [[1,6],[2],[3],[4],[5]]
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [[1,7],[2],[3],[4],[5],[6]]
=> 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7]]
=> 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [[1,7],[2],[3],[4],[5],[6]]
=> 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [7,6,4,5,3,2,1] => [[1,5],[2],[3],[4],[6],[7]]
=> 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => [[1,6],[2],[3],[4],[5],[7]]
=> 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [[1,7],[2],[3],[4],[5],[6]]
=> 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6],[7]]
=> 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => [[1,5],[2],[3],[4],[6],[7]]
=> 3
Description
The charge of a standard tableau.
Matching statistic: St000010
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(load all 3 compositions to match this statistic)
Mp00020: Binary trees āto Tamari-corresponding Dyck pathā¶ Dyck paths
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> []
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> []
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 3
Description
The length of the partition.
Matching statistic: St000012
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00018: Binary trees āleft border symmetryā¶ Binary trees
Mp00012: Binary trees āto Dyck path: up step, left tree, down step, right treeā¶ Dyck paths
St000012: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00012: Binary trees āto Dyck path: up step, left tree, down step, right treeā¶ Dyck paths
St000012: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [1,0]
=> 0
[.,[.,.]]
=> [.,[.,.]]
=> [1,0,1,0]
=> 0
[[.,.],.]
=> [[.,.],.]
=> [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[[.,[.,.]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4
[[.,[.,[.,.]]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> 3
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000160
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00020: Binary trees āto Tamari-corresponding Dyck pathā¶ Dyck paths
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St000160: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St000160: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> []
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> []
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 3
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000394
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00018: Binary trees āleft border symmetryā¶ Binary trees
Mp00012: Binary trees āto Dyck path: up step, left tree, down step, right treeā¶ Dyck paths
St000394: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00012: Binary trees āto Dyck path: up step, left tree, down step, right treeā¶ Dyck paths
St000394: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [1,0]
=> 0
[.,[.,.]]
=> [.,[.,.]]
=> [1,0,1,0]
=> 0
[[.,.],.]
=> [[.,.],.]
=> [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[[.,[.,.]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4
[[.,[.,[.,.]]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> 3
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000548
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00020: Binary trees āto Tamari-corresponding Dyck pathā¶ Dyck paths
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St000548: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St000548: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> []
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> []
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 3
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St000032
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00018: Binary trees āleft border symmetryā¶ Binary trees
Mp00012: Binary trees āto Dyck path: up step, left tree, down step, right treeā¶ Dyck paths
St000032: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00012: Binary trees āto Dyck path: up step, left tree, down step, right treeā¶ Dyck paths
St000032: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[[.,.],.]
=> [[.,.],.]
=> [1,1,0,0]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[[.,[.,.]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[.,[.,[.,[.,.]]]],[.,.]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
Description
The number of elements smaller than the given Dyck path in the Tamari Order.
Matching statistic: St000008
Mp00020: Binary trees āto Tamari-corresponding Dyck pathā¶ Dyck paths
Mp00102: Dyck paths ārise compositionā¶ Integer compositions
Mp00038: Integer compositions āreverseā¶ Integer compositions
St000008: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00102: Dyck paths ārise compositionā¶ Integer compositions
Mp00038: Integer compositions āreverseā¶ Integer compositions
St000008: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [1] => 0
[.,[.,.]]
=> [1,1,0,0]
=> [2] => [2] => 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [3] => [3] => 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [2,1] => [1,2] => 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,1] => [1,2] => 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4] => 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 3
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,4] => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [2,3] => 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => 3
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [2,3] => 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,4] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => 2
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,4] => 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => [6] => 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,1] => [1,5] => 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,2] => [2,4] => 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1] => [1,5] => 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3] => [3,3] => 3
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,2] => [2,4] => 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1] => [1,5] => 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => [4,2] => 4
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3] => [3,3] => 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => [2,4] => 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1] => [1,5] => 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [5,1] => 5
[[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => [4,2] => 4
[[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => [3,3] => 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => [2,4] => 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [1,5] => 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => [7] => 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,1] => [1,6] => 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,2] => [2,5] => 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,1] => [1,6] => 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,3] => [3,4] => 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,2] => [2,5] => 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,1] => [1,6] => 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4] => [4,3] => 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3] => [3,4] => 3
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000024
Mp00016: Binary trees āleft-right symmetryā¶ Binary trees
Mp00014: Binary trees āto 132-avoiding permutationā¶ Permutations
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
St000024: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00014: Binary trees āto 132-avoiding permutationā¶ Permutations
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
St000024: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [1] => [1,0]
=> 0
[.,[.,.]]
=> [[.,.],.]
=> [1,2] => [1,0,1,0]
=> 0
[[.,.],.]
=> [.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 2
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [[[[[.,[.,.]],.],.],.],.]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [[[[.,[[.,.],.]],.],.],.]
=> [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [[[[[.,.],.],[.,.]],.],.]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[.,.],[[.,.],.]],.],.]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [[[.,[[[.,.],.],.]],.],.]
=> [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],[.,.]],.]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4
[.,[[.,[.,.]],[.,[.,.]]]]
=> [[[[.,.],.],[[.,.],.]],.]
=> [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [[[.,.],[[[.,.],.],.]],.]
=> [3,4,5,1,2,6] => [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [[.,[[[[.,.],.],.],.]],.]
=> [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[[.,[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[[.,[.,[.,.]]],[.,[.,.]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [[.,.],[[[[.,.],.],.],.]]
=> [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],.],.],.]]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[[[[[.,.],.],[.,.]],.],.],.]
=> [4,1,2,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[[[[.,.],[[.,.],.]],.],.],.]
=> [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [[[[.,[[[.,.],.],.]],.],.],.]
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],[.,.]],.],.]
=> [5,1,2,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> [4,5,1,2,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> 3
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000053
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00020: Binary trees āto Tamari-corresponding Dyck pathā¶ Dyck paths
Mp00122: Dyck paths āElizalde-Deutsch bijectionā¶ Dyck paths
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
St000053: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00122: Dyck paths āElizalde-Deutsch bijectionā¶ Dyck paths
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
St000053: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 4
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 3
Description
The number of valleys of the Dyck path.
The following 354 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000147The largest part of an integer partition. St000169The cocharge of a standard tableau. St000211The rank of the set partition. St000330The (standard) major index of a standard tableau. St000378The diagonal inversion number of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000692Babson and SteingrĆmsson's statistic of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000984The number of boxes below precisely one peak. St000996The number of exclusive left-to-right maxima of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001094The depth index of a set partition. St001161The major index north count of a Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001280The number of parts of an integer partition that are at least two. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001697The shifted natural comajor index of a standard Young tableau. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000058The order of a permutation. St000381The largest part of an integer composition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000808The number of up steps of the associated bargraph. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama 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:
St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St000439The position of the first down step of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000947The major index east count of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001721The degree of a binary word. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St000209Maximum difference of elements in cycles. St000374The number of exclusive right-to-left minima of a permutation. St000391The sum of the positions of the ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000446The disorder of a permutation. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000503The maximal difference between two elements in a common block. St000693The modular (standard) major index of a standard tableau. St000703The number of deficiencies of a permutation. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000946The sum of the skew hook positions in a Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000678The number of up steps after the last double rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000809The reduced reflection length of the permutation. St000833The comajor index of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000662The staircase size of the code of a permutation. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000054The first entry of the permutation. St000306The bounce count of a Dyck path. St000797The stat`` of a permutation. St000798The makl of a permutation. St001933The largest multiplicity of a part in an integer partition. St000288The number of ones in a binary word. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000627The exponent of a binary word. St000667The greatest common divisor of the parts of the partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001267The length of the Lyndon factorization of the binary word. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001485The modular major index of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001884The number of borders of a binary word. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000518The number of distinct subsequences in a binary word. St000519The largest length of a factor maximising the subword complexity. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000161The sum of the sizes of the right subtrees of a binary tree. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000733The row containing the largest entry of a standard tableau. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001571The Cartan determinant of the integer partition. St001671Haglund's hag of a permutation. St000041The number of nestings of a perfect matching. St000653The last descent of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001489The maximum of the number of descents and the number of inverse descents. St000110The number of permutations less than or equal to a permutation in left weak order. St000470The number of runs in a permutation. St000237The number of small exceedances. St000354The number of recoils of a permutation. St000651The maximal size of a rise in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000007The number of saliances of the permutation. St000993The multiplicity of the largest part of an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St001128The exponens consonantiae of a partition. St001184Number of indecomposable injective modules with grade at least 1 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000246The number of non-inversions of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001268The size of the largest ordinal summand in the poset. St000864The number of circled entries of the shifted recording tableau of a permutation. 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. St000542The number of left-to-right-minima of a permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St000304The load of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000339The maf index of a permutation. St000796The stat' of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001239The largest vector space dimension of the double dual of a simple module in the corresponding 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. St000004The major index of a permutation. St000006The dinv of a Dyck path. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000156The Denert index of a permutation. St000204The number of internal nodes of a binary tree. St000224The sorting index of a permutation. St000331The number of upper interactions of a Dyck path. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000356The number of occurrences of the pattern 13-2. 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. St001077The prefix exchange distance of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the 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. 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 $eA$. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001718The number of non-empty open intervals in a poset. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000472The sum of the ascent bottoms of a permutation. St001480The number of simple summands of the module J^2/J^3. St000082The number of elements smaller than a binary tree in Tamari order. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St001346The number of parking functions that give the same permutation. St000216The absolute length of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000652The maximal difference between successive positions of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000461The rix statistic of a permutation. St000710The number of big deficiencies of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001152The number of pairs with even minimum in a perfect matching. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001684The reduced word complexity of a permutation. St001727The number of invisible inversions of a permutation. 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. St000327The number of cover relations in a poset. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. St001864The number of excedances of a signed permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001961The sum of the greatest common divisors of all pairs of parts. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000284The Plancherel distribution on integer partitions. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001568The smallest positive integer that does not appear twice in the partition. St000455The second largest eigenvalue of a graph if it is integral. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001209The pmaj statistic of a parking function. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000052The number of valleys of a Dyck path not on the x-axis. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000259The diameter of a connected graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000562The number of internal points of a set partition. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000711The number of big exceedences of a permutation. St001728The number of invisible descents of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000254The nesting number of a set partition. St000570The Edelman-Greene number of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000942The number of critical left to right maxima of the parking functions. St001645The pebbling number of a connected graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001863The number of weak excedances of a signed permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000256The number of parts from which one can substract 2 and still get an integer partition.
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