searching the database
Your data matches 211 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000783
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St000783: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000783: 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
[1,0,1,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1 = 2 - 1
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St001176
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
St001176: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00204: Permutations āLLPSā¶ Integer partitions
St001176: Integer partitions 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
[1,0,1,0]
=> [1,2] => [1,1]
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [4,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 1 = 2 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
Mp00062: Permutations āLehmer-code to major-code bijectionā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00062: Permutations āLehmer-code to major-code bijectionā¶ Permutations
Mp00204: Permutations āLLPSā¶ 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]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [2,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [2,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [2,2]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,4,3,2] => [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [2,1,1,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,5,4,1,3] => [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [4,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [2,1,1,1]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => [2,2,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,1,4,3] => [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => [4,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,1,1]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,1,5,3,4] => [2,2,1]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [2,1,4,3,5] => [2,2,1]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,2,1]
=> 3
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,5,2,4,3] => [3,1,1]
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [1,4,3,2,5] => [3,1,1]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [1,3,2,5,4] => [2,2,1]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => [3,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5,1,4,3,2] => [4,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,5,4] => [3,2]
=> 2
Description
The length of the partition.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000011: 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
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000011: 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]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000013
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00132: Dyck paths āswitch returns and last double riseā¶ Dyck paths
St000013: 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
Mp00132: Dyck paths āswitch returns and last double riseā¶ Dyck paths
St000013: 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]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St001432
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00143: Dyck paths āinverse promotionā¶ Dyck paths
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St001432: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00143: Dyck paths āinverse promotionā¶ Dyck paths
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St001432: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> 3
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> 2
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St000228
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000228: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000228: Integer partitions 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
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2]
=> [2]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2]
=> [2]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> 1 = 2 - 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000394
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000394: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ 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]
=> [1] => [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000459
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000459: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000459: Integer partitions 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
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2]
=> [2]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2]
=> [2]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> 1 = 2 - 1
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000662
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000662: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000662: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> []
=> [] => 0 = 1 - 1
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> []
=> []
=> [] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> []
=> []
=> [] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 2 - 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
The following 201 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000738The first entry in the last row of a standard tableau. St000288The number of ones in a binary word. St000377The dinv defect of an integer partition. St000507The number of ascents of a standard tableau. St000157The number of descents of a standard tableau. St000306The bounce count of a Dyck path. 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. St000144The pyramid weight of the Dyck path. St000331The number of upper interactions of a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000460The hook length of the last cell along the main diagonal of an integer partition. St000527The width of the poset. St000758The length of the longest staircase fitting into an integer composition. St000829The Ulam distance of a permutation to the identity permutation. St000862The number of parts of the shifted shape of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001298The number of repeated entries in the Lehmer code of a permutation. St001380The number of monomer-dimer tilings of a Ferrers diagram. St000439The position of the first down step of a Dyck path. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000632The jump number of the poset. St000676The number of odd rises of a Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000019The cardinality of the support of a permutation. St000214The number of adjacencies of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000470The number of runs in a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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:
St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001277The degeneracy of a graph. St001489The maximum of the number of descents and the number of inverse descents. 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. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000678The number of up steps after the last double rise of a Dyck path. St000912The number of maximal antichains in a poset. St000018The number of inversions of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St000354The number of recoils of a permutation. St000441The number of successions of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000494The number of inversions of distance at most 3 of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000336The leg major index of a standard tableau. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000240The number of indices that are not small excedances. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000991The number of right-to-left minima of a permutation. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001530The depth of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000067The inversion number of the alternating sign matrix. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000332The positive inversions of an alternating sign matrix. St000653The last descent 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. St001175The size of a partition minus the hook length of the base cell. 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 $eA$. 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. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001427The number of descents of a signed permutation. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001812The biclique partition number of a graph. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000291The number of descents of a binary word. St001321The number of vertices of the largest induced subforest of a graph. St000996The number of exclusive left-to-right maxima of a permutation. St000731The number of double exceedences of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000619The number of cyclic descents of a permutation. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000837The number of ascents of distance 2 of a permutation. St000836The number of descents of distance 2 of a permutation. St000314The number of left-to-right-maxima of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001726The number of visible inversions of a permutation. St000039The number of crossings of a permutation. St000185The weighted size of a partition. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001214The aft of an integer partition. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000145The Dyson rank of a partition. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001668The number of points of the poset minus the width of the poset. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001863The number of weak excedances of a signed permutation. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. 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. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000493The los statistic of a set partition. St000497The lcb statistic of a set partition. St001651The Frankl number of a lattice. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000028The number of stack-sorts needed to sort a permutation. St000352The Elizalde-Pak rank of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001435The number of missing boxes in the first row. St001488The number of corners of a skew partition. St001626The number of maximal proper sublattices of a lattice. St000834The number of right outer peaks of a permutation. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000568The hook number of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000098The chromatic number of a graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000035The number of left outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation. St000264The girth of a graph, which is not a tree. St000366The number of double descents of a permutation. St000451The length of the longest pattern of the form k 1 2. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St001581The achromatic number of a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database ā it's very simple and we need your support!