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Your data matches 30 different statistics following compositions of up to 3 maps.
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Matching statistic: St000844
(load all 1278 compositions to match this statistic)
(load all 1278 compositions to match this statistic)
St000844: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 = 0 + 1
[2,1] => 2 = 1 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 3 = 2 + 1
[3,1,2] => 3 = 2 + 1
[3,2,1] => 3 = 2 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => 3 = 2 + 1
[1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => 3 = 2 + 1
[2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 3 = 2 + 1
[2,3,4,1] => 4 = 3 + 1
[2,4,1,3] => 4 = 3 + 1
[2,4,3,1] => 4 = 3 + 1
[3,1,2,4] => 3 = 2 + 1
[3,1,4,2] => 4 = 3 + 1
[3,2,1,4] => 3 = 2 + 1
[3,2,4,1] => 4 = 3 + 1
[3,4,1,2] => 4 = 3 + 1
[3,4,2,1] => 4 = 3 + 1
[4,1,2,3] => 4 = 3 + 1
[4,1,3,2] => 4 = 3 + 1
[4,2,1,3] => 4 = 3 + 1
[4,2,3,1] => 4 = 3 + 1
[4,3,1,2] => 4 = 3 + 1
[4,3,2,1] => 4 = 3 + 1
[1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => 3 = 2 + 1
[1,2,5,3,4] => 3 = 2 + 1
[1,2,5,4,3] => 3 = 2 + 1
[1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => 2 = 1 + 1
[1,3,4,2,5] => 3 = 2 + 1
[1,3,4,5,2] => 4 = 3 + 1
[1,3,5,2,4] => 4 = 3 + 1
[1,3,5,4,2] => 4 = 3 + 1
[1,4,2,3,5] => 3 = 2 + 1
[1,4,2,5,3] => 4 = 3 + 1
[1,4,3,2,5] => 3 = 2 + 1
[1,4,3,5,2] => 4 = 3 + 1
[1,4,5,2,3] => 4 = 3 + 1
[1,4,5,3,2] => 4 = 3 + 1
Description
The size of the largest block in the direct sum decomposition of a permutation.
A component of a permutation $\pi$ is a set of consecutive numbers $\{a,a+1,\dots, b\}$ such that $a\leq \pi(i) \leq b$ for all $a\leq i\leq b$.
This statistic is the size of the largest component which does not properly contain another component.
Matching statistic: St000209
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 2
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 3
Description
Maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the maximum of this value over all cycles in the permutation.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => 0 => 0
[2,1] => 0 => 1 => 1
[1,2,3] => 11 => 00 => 0
[1,3,2] => 10 => 01 => 1
[2,1,3] => 01 => 10 => 1
[2,3,1] => 00 => 11 => 2
[3,1,2] => 00 => 11 => 2
[3,2,1] => 00 => 11 => 2
[1,2,3,4] => 111 => 000 => 0
[1,2,4,3] => 110 => 001 => 1
[1,3,2,4] => 101 => 010 => 1
[1,3,4,2] => 100 => 011 => 2
[1,4,2,3] => 100 => 011 => 2
[1,4,3,2] => 100 => 011 => 2
[2,1,3,4] => 011 => 100 => 1
[2,1,4,3] => 010 => 101 => 1
[2,3,1,4] => 001 => 110 => 2
[2,3,4,1] => 000 => 111 => 3
[2,4,1,3] => 000 => 111 => 3
[2,4,3,1] => 000 => 111 => 3
[3,1,2,4] => 001 => 110 => 2
[3,1,4,2] => 000 => 111 => 3
[3,2,1,4] => 001 => 110 => 2
[3,2,4,1] => 000 => 111 => 3
[3,4,1,2] => 000 => 111 => 3
[3,4,2,1] => 000 => 111 => 3
[4,1,2,3] => 000 => 111 => 3
[4,1,3,2] => 000 => 111 => 3
[4,2,1,3] => 000 => 111 => 3
[4,2,3,1] => 000 => 111 => 3
[4,3,1,2] => 000 => 111 => 3
[4,3,2,1] => 000 => 111 => 3
[1,2,3,4,5] => 1111 => 0000 => 0
[1,2,3,5,4] => 1110 => 0001 => 1
[1,2,4,3,5] => 1101 => 0010 => 1
[1,2,4,5,3] => 1100 => 0011 => 2
[1,2,5,3,4] => 1100 => 0011 => 2
[1,2,5,4,3] => 1100 => 0011 => 2
[1,3,2,4,5] => 1011 => 0100 => 1
[1,3,2,5,4] => 1010 => 0101 => 1
[1,3,4,2,5] => 1001 => 0110 => 2
[1,3,4,5,2] => 1000 => 0111 => 3
[1,3,5,2,4] => 1000 => 0111 => 3
[1,3,5,4,2] => 1000 => 0111 => 3
[1,4,2,3,5] => 1001 => 0110 => 2
[1,4,2,5,3] => 1000 => 0111 => 3
[1,4,3,2,5] => 1001 => 0110 => 2
[1,4,3,5,2] => 1000 => 0111 => 3
[1,4,5,2,3] => 1000 => 0111 => 3
[1,4,5,3,2] => 1000 => 0111 => 3
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000503
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> {{1},{2}}
=> 0
[2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
Description
The maximal difference between two elements in a common block.
Matching statistic: St000956
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 2
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 3
Description
The maximal displacement of a permutation.
This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$.
This statistic without the absolute value is the maximal drop size [[St000141]].
Matching statistic: St001120
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
[1,2] => ([],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => ([],5)
=> ([],5)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The length of a longest path in a graph.
Matching statistic: St000147
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => ([],2)
=> [1,1]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3 = 2 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 4 = 3 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
Description
The largest part of an integer partition.
Matching statistic: St000381
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [2] => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [3] => 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3] => 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [4] => 4 = 3 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [4] => 4 = 3 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4] => 4 = 3 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [4] => 4 = 3 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [4] => 4 = 3 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4] => 4 = 3 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4] => 4 = 3 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4 = 3 + 1
Description
The largest part of an integer composition.
Matching statistic: St000171
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,2] => ([],2)
=> 0
[2,1] => [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000442
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 0
[2,1] => 0 => [2] => [1,1,0,0]
=> 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,5,3,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
Description
The maximal area to the right of an up step of a Dyck path.
The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000013The height of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001268The size of the largest ordinal summand in the poset. St000662The staircase size of the code of a permutation. St001644The dimension of a graph. St001090The number of pop-stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St001330The hat guessing number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000454The largest eigenvalue of a graph if it is integral. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001645The pebbling number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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