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Your data matches 447 different statistics following compositions of up to 3 maps.
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Matching statistic: St000026
(load all 76 compositions to match this statistic)
(load all 76 compositions to match this statistic)
Mp00007: Alternating sign matrices āto Dyck pathā¶ Dyck paths
St000026: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000026: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
Description
The position of the first return of a Dyck path.
Matching statistic: St000019
(load all 259 compositions to match this statistic)
(load all 259 compositions to match this statistic)
Mp00002: Alternating sign matrices āto left key permutationā¶ Permutations
St000019: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000019: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0 = 1 - 1
[[0,1],[1,0]]
=> [2,1] => 1 = 2 - 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 2 = 3 - 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [2,3,1] => 2 = 3 - 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 2 = 3 - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => 3 = 4 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => 3 = 4 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [4,2,1,3] => 3 = 4 - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => 3 = 4 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => 3 = 4 - 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => 3 = 4 - 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => 3 = 4 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => 3 = 4 - 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [3,4,2,1] => 3 = 4 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 3 = 4 - 1
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [3,5,1,2,4] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [3,5,1,2,4] => 4 = 5 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [5,3,1,2,4] => 4 = 5 - 1
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [2,5,3,1,4] => 4 = 5 - 1
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [2,5,3,1,4] => 4 = 5 - 1
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [2,5,3,1,4] => 4 = 5 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [5,2,3,1,4] => 4 = 5 - 1
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [3,5,2,1,4] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [3,5,2,1,4] => 4 = 5 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [5,3,2,1,4] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [4,5,1,2,3] => 4 = 5 - 1
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [5,4,1,2,3] => 4 = 5 - 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,4,5,1,3] => 4 = 5 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,4,5,1,3] => 4 = 5 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [4,2,5,1,3] => 4 = 5 - 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,4,5,1,3] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [4,2,5,1,3] => 4 = 5 - 1
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => 4 = 5 - 1
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => 4 = 5 - 1
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => 4 = 5 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [5,2,4,1,3] => 4 = 5 - 1
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => 4 = 5 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [5,2,4,1,3] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [4,5,2,1,3] => 4 = 5 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [5,4,2,1,3] => 4 = 5 - 1
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [3,5,1,4,2] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [3,5,1,4,2] => 4 = 5 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [5,3,1,4,2] => 4 = 5 - 1
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [3,5,1,4,2] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [3,5,1,4,2] => 4 = 5 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [5,3,1,4,2] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [4,5,1,3,2] => 4 = 5 - 1
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [5,4,1,3,2] => 4 = 5 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [4,5,1,3,2] => 4 = 5 - 1
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000228
(load all 116 compositions to match this statistic)
(load all 116 compositions to match this statistic)
Mp00002: Alternating sign matrices āto left key permutationā¶ Permutations
Mp00060: Permutations āRobinson-Schensted tableau shapeā¶ Integer partitions
St000228: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00060: Permutations āRobinson-Schensted tableau shapeā¶ Integer partitions
St000228: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1]
=> 1
[[0,1],[1,0]]
=> [2,1] => [1,1]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [2,1]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [2,3,1] => [2,1]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,1,1]
=> 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [2,2]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [2,2]
=> 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [4,2,1,3] => [2,1,1]
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => [2,2]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [2,1,1]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [2,1,1]
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [2,1,1]
=> 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => [2,1,1]
=> 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [3,4,2,1] => [2,1,1]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,1,1,1]
=> 4
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [3,5,1,2,4] => [3,2]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [3,5,1,2,4] => [3,2]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [5,3,1,2,4] => [3,1,1]
=> 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [2,5,3,1,4] => [3,1,1]
=> 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [2,5,3,1,4] => [3,1,1]
=> 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [2,5,3,1,4] => [3,1,1]
=> 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [5,2,3,1,4] => [3,1,1]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [3,5,2,1,4] => [2,2,1]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [3,5,2,1,4] => [2,2,1]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [5,3,2,1,4] => [2,1,1,1]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [4,5,1,2,3] => [3,2]
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [5,4,1,2,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,4,5,1,3] => [3,2]
=> 5
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,4,5,1,3] => [3,2]
=> 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [4,2,5,1,3] => [2,2,1]
=> 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,4,5,1,3] => [3,2]
=> 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [4,2,5,1,3] => [2,2,1]
=> 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => [2,2,1]
=> 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => [2,2,1]
=> 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => [2,2,1]
=> 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [5,2,4,1,3] => [2,2,1]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => [2,2,1]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [2,5,4,1,3] => [2,2,1]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [5,2,4,1,3] => [2,2,1]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [4,5,2,1,3] => [2,2,1]
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [5,4,2,1,3] => [2,1,1,1]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [3,5,1,4,2] => [2,2,1]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [3,5,1,4,2] => [2,2,1]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [5,3,1,4,2] => [2,2,1]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [3,5,1,4,2] => [2,2,1]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [3,5,1,4,2] => [2,2,1]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [5,3,1,4,2] => [2,2,1]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [4,5,1,3,2] => [2,2,1]
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [5,4,1,3,2] => [2,1,1,1]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [4,5,1,3,2] => [2,2,1]
=> 5
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000381
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices āto Dyck pathā¶ Dyck paths
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000381: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000381: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1] => 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> [2] => 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [3] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
Description
The largest part of an integer composition.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices āto Dyck pathā¶ Dyck paths
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1] => 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> [2] => 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [3] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices āto Dyck pathā¶ Dyck paths
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1] => 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> [2] => 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [3] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
Description
The last part of an integer composition.
Matching statistic: St000395
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00007: Alternating sign matrices āto Dyck pathā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
St000395: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
St000395: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1,0]
=> 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000505
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00007: Alternating sign matrices āto Dyck pathā¶ Dyck paths
Mp00138: Dyck paths āto noncrossing partitionā¶ Set partitions
St000505: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00138: Dyck paths āto noncrossing partitionā¶ Set partitions
St000505: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> {{1}}
=> 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> {{1,2}}
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 5
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
Description
The biggest entry in the block containing the 1.
Matching statistic: St000657
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices āto Dyck pathā¶ Dyck paths
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000657: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000657: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1] => 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> [2] => 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [3] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
Description
The smallest part of an integer composition.
Matching statistic: St000740
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00007: Alternating sign matrices āto Dyck pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000740: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000740: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1] => 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> [1,2] => 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [2,1,3] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 5
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 5
[[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 5
[[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 5
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 5
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
The following 437 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000808The number of up steps of the associated bargraph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001268The size of the largest ordinal summand in the poset. St001497The position of the largest weak excedence of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001778The largest greatest common divisor of an element and its image in a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001332The number of steps on the non-negative side of the walk associated with the permutation. 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. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000054The first entry of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000215The number of adjacencies of a permutation, zero appended. St000258The burning number of a graph. St000273The domination number of a graph. St000288The number of ones in a binary word. St000293The number of inversions of a binary word. St000336The leg major index of a standard tableau. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. 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. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000503The maximal difference between two elements in a common block. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000543The size of the conjugacy class of a binary word. St000544The cop number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000636The hull number of a graph. St000667The greatest common divisor of the parts of the partition. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000839The largest opener of a set partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000883The number of longest increasing subsequences of a permutation. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001267The length of the Lyndon factorization of the binary word. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001430The number of positive entries in a signed permutation. St001437The flex of a binary word. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001614The cyclic permutation representation number of a skew partition. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001691The number of kings in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000010The length of the partition. St000053The number of valleys of the Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000081The number of edges of a graph. St000108The number of partitions contained in the given partition. St000141The maximum drop size of a permutation. St000145The Dyson rank of a partition. St000171The degree of the graph. St000214The number of adjacencies of a permutation. St000246The number of non-inversions of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000519The largest length of a factor maximising the subword complexity. St000532The total number of rook placements on a Ferrers board. St000546The number of global descents of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000778The metric dimension of a graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. 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. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001096The size of the overlap set of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001279The sum of the 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. St001340The cardinality of a minimal non-edge isolating set of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001479The number of bridges of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001658The total number of rook placements on a Ferrers board. St001777The number of weak descents in an integer composition. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St001958The degree of the polynomial interpolating the values of a permutation. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000501The size of the first part in the decomposition of a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St000031The number of cycles in the cycle decomposition of a permutation. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St000240The number of indices that are not small excedances. St000294The number of distinct factors of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000518The number of distinct subsequences in a binary word. St000703The number of deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001622The number of join-irreducible elements of a lattice. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000209Maximum difference of elements in cycles. St000296The length of the symmetric border of a binary word. St000627The exponent of a binary word. St000806The semiperimeter of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001371The length of the longest Yamanouchi prefix of a binary word. 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. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000015The number of peaks of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000087The number of induced subgraphs. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000325The width of the tree associated to a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000470The number of runs in a permutation. St000477The weight of a partition according to Alladi. St000542The number of left-to-right-minima of a permutation. St000547The number of even non-empty partial sums of an integer partition. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000668The least common multiple of the parts of the partition. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000708The product of the parts of an integer partition. St000733The row containing the largest entry of a standard tableau. St000770The major index of an integer partition when read from bottom to top. St000863The length of the first row of the shifted shape of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000996The number of exclusive left-to-right maxima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001052The length of the exterior of a permutation. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001252Half the sum of the even parts of a partition. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001725The harmonious chromatic number of a graph. St001828The Euler characteristic of a graph. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000058The order of a permutation. St000133The "bounce" of a permutation. St000167The number of leaves of an ordered tree. St000238The number of indices that are not small weak excedances. St000297The number of leading ones in a binary word. St000304The load of a permutation. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000451The length of the longest pattern of the form k 1 2. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000653The last descent of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000878The number of ones minus the number of zeros of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St000983The length of the longest alternating subword. 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. St001199The dominant 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. 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. 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001405The number of bonds in a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001721The degree of a binary word. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001925The minimal number of zeros in a row of an alternating sign matrix. St000656The number of cuts of a poset. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000837The number of ascents of distance 2 of a permutation. St000874The position of the last double rise in a Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001480The number of simple summands of the module J^2/J^3. St000673The number of non-fixed points of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000064The number of one-box pattern of a permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000471The sum of the ascent tops of a permutation. St000625The sum of the minimal distances to a greater element. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000990The first ascent of a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St000060The greater neighbor of the maximum. St000197The number of entries equal to positive one in the alternating sign matrix. St000354The number of recoils of a permutation. St000520The number of patterns in a permutation. St000530The number of permutations with the same descent word as the given permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St000836The number of descents of distance 2 of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000924The number of topologically connected components of a perfect matching. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000840The number of closers smaller than the largest opener in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000672The number of minimal elements in Bruhat order not less than the permutation. St001917The order of toric promotion on the set of labellings of a graph. St000018The number of inversions of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000245The number of ascents of a permutation. St000146The Andrews-Garvan crank of a partition. St000157The number of descents of a standard tableau. St000507The number of ascents of a standard tableau. St000035The number of left outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000834The number of right outer peaks of a permutation. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001468The smallest fixpoint of a permutation. St000029The depth of a permutation. 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. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000780The size of the orbit under rotation of a perfect matching. St000809The reduced reflection length of the permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001429The number of negative entries in a signed permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the 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. St000235The number of indices that are not cyclical small weak excedances. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001834The number of non-isomorphic minors of a graph. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001706The number of closed sets in a graph. St001557The number of inversions of the second entry of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000327The number of cover relations in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001948The number of augmented double ascents of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001927Sparre Andersen's number of positives of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000050The depth or height of a binary tree. St000356The number of occurrences of the pattern 13-2. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000945The number of matchings in the dihedral orbit of a perfect matching. St001651The Frankl number of a lattice. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001280The number of parts of an integer partition that are at least two. St001432The order dimension of the partition. St001434The number of negative sum pairs of a signed permutation. St000264The girth of a graph, which is not a tree. St000884The number of isolated descents of a permutation.
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