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Your data matches 164 different statistics following compositions of up to 3 maps.
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Matching statistic: St000784
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> 0
[1,0,1,0]
=> [1]
=> 1
[1,1,0,0]
=> []
=> 0
[1,0,1,0,1,0]
=> [2,1]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,0,0,1,0]
=> [2]
=> 2
[1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St000209
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,4,2,1] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,3,2,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,5,3,1] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,5,4,3,1] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,4,1,3] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,5,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,4,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,5,4,2,1] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,1,2] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [3,5,1,2,4] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [3,1,2,4,5] => 2
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: St000956
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 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: St000734
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [[1]]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [[1,2]]
=> 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,3],[2],[4]]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,4],[3],[5]]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,5],[3],[4]]
=> 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[1,3,4],[2],[5]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[1,3,5],[2],[4]]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[1,3,5],[2],[4]]
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [[1,3,4],[2],[5]]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[1,3],[2],[4],[5]]
=> 3 = 2 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000019
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,3,1] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [2,4,1,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [4,1,3,2] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,2,4,1] => [4,2,1,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => [4,2,3,1] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,4,1,2] => [1,4,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [2,3,5,1,4] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [3,4,2,1,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [3,5,1,4,2] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,3,5,2] => [3,5,2,1,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => [3,5,2,4,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [2,5,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,5,2,3] => [2,1,5,3,4] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,5,1] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [3,2,4,1,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,3,5,2,1] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [3,2,5,1,4] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,5,1] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,1,3,2,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,1,2,4,3] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,2,3,5,1] => [5,1,3,2,4] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,2,4,1,5] => [4,2,1,3,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,2,5,4,1] => [5,3,2,4,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,2,4,5,1] => [5,2,1,3,4] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,2,5,1,3] => [1,4,5,2,3] => 3
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000141
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [3,2,1] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [4,3,2,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [4,2,3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [4,3,2,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [4,2,3,1] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [4,3,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [5,2,3,4,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [4,2,3,1,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => [5,4,3,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [5,4,3,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [4,5,3,1,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [5,4,3,2,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => [5,4,3,2,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [4,3,2,1,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => [5,4,3,2,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => [5,4,3,2,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [5,4,3,2,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [3,5,1,4,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [3,4,1,2,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [4,5,3,1,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [5,4,3,2,1] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [4,2,5,1,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [5,2,3,4,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [5,4,3,2,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [5,3,2,4,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,5,3,4,1] => [5,4,3,2,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [5,4,3,2,1] => 4
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000171
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ 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
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => ([],2)
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000316
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [3,2,1] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [4,3,2,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [4,2,3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [4,3,2,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [4,2,3,1] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [4,3,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [5,2,3,4,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [4,2,3,1,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => [5,4,3,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [5,4,3,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [4,5,3,1,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [5,4,3,2,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => [5,4,3,2,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [4,3,2,1,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => [5,4,3,2,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => [5,4,3,2,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [5,4,3,2,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [3,5,1,4,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [3,4,1,2,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [4,5,3,1,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [5,4,3,2,1] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [4,2,5,1,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [5,2,3,4,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [5,4,3,2,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [5,3,2,4,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,5,3,4,1] => [5,4,3,2,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [5,4,3,2,1] => 4
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St000653
(load all 62 compositions to match this statistic)
(load all 62 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,3,1,2,5] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,1,5] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [4,1,3,2,5] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,4,1,2,5] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,4,2,3,5] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [5,4,3,2,1,6] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [5,4,3,1,2,6] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => [5,4,2,3,1,6] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => [5,4,1,3,2,6] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [5,4,1,2,3,6] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => [5,3,4,2,1,6] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => [5,3,4,1,2,6] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => [5,2,4,3,1,6] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => [5,1,4,3,2,6] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => [5,1,4,2,3,6] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => [5,2,3,4,1,6] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => [5,1,3,4,2,6] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => [5,1,3,2,4,6] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [5,1,2,3,4,6] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [4,5,3,2,1,6] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => [4,5,3,1,2,6] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => [4,5,2,3,1,6] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,6,4,5,1] => [4,5,1,3,2,6] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => [4,5,1,2,3,6] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => [3,5,4,2,1,6] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => [3,5,4,1,2,6] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => [2,5,4,3,1,6] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => [1,5,4,3,2,6] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,5,4,2,3,6] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => [2,5,3,4,1,6] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,5,3,4,2,6] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => [1,5,3,2,4,6] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => [1,5,2,3,4,6] => 2
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St001742
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001742: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001742: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => ([],2)
=> 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
Description
The difference of the maximal and the minimal degree in a graph.
The graph is regular if and only if this statistic is zero.
The following 154 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000054The first entry of the permutation. St000240The number of indices that are not small excedances. St000505The biggest entry in the block containing the 1. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001725The harmonious chromatic number of a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000503The maximal difference between two elements in a common block. St001721The degree of a binary word. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001566The length of the longest arithmetic progression in a permutation. St001645The pebbling number of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000259The diameter of a connected graph. St000133The "bounce" of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001246The maximal difference between two consecutive entries of a permutation. St000060The greater neighbor of the maximum. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000235The number of indices that are not cyclical small weak excedances. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001555The order of a signed permutation. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000662The staircase size of the code of a permutation. St001863The number of weak excedances of a signed permutation. St000422The energy of a graph, if it is integral. St001090The number of pop-stack-sorts needed to sort a permutation. St000356The number of occurrences of the pattern 13-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000742The number of big ascents of a permutation after prepending zero. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000670The reversal length of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000667The greatest common divisor of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000731The number of double exceedences of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001875The number of simple modules with projective dimension at most 1. St001894The depth of a signed permutation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000451The length of the longest pattern of the form k 1 2. St000624The normalized sum of the minimal distances to a greater element. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000155The number of exceedances (also excedences) of a permutation. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000358The number of occurrences of the pattern 31-2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000491The number of inversions of a set partition. St000516The number of stretching pairs of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000682The Grundy value of Welter's game on a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000837The number of ascents of distance 2 of a permutation. St001115The number of even descents of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001569The maximal modular displacement of a permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001801Half the number of preimage-image pairs of different parity in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001822The number of alignments of a signed permutation. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000092The number of outer peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000619The number of cyclic descents of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001557The number of inversions of the second entry of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001778The largest greatest common divisor of an element and its image in a permutation. St001792The arboricity of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000735The last entry on the main diagonal of a standard tableau. St000632The jump number of the poset. St001638The book thickness of a graph. St000307The number of rowmotion orbits of a poset. St000652The maximal difference between successive positions of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001926Sparre Andersen's position of the maximum of a signed permutation. St001623The number of doubly irreducible elements of a lattice.
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