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Matching statistic: St000056
(load all 1278 compositions to match this statistic)
(load all 1278 compositions to match this statistic)
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 2
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 3
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 3
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 3
[1,4,2,5,3] => 2
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
Matching statistic: St000234
(load all 1278 compositions to match this statistic)
(load all 1278 compositions to match this statistic)
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 1 = 2 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 2 = 3 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 0 = 1 - 1
[3,1,2] => 0 = 1 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 3 = 4 - 1
[1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 0 = 1 - 1
[2,4,1,3] => 0 = 1 - 1
[2,4,3,1] => 0 = 1 - 1
[3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => 0 = 1 - 1
[3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => 0 = 1 - 1
[3,4,1,2] => 0 = 1 - 1
[3,4,2,1] => 0 = 1 - 1
[4,1,2,3] => 0 = 1 - 1
[4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => 0 = 1 - 1
[4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 4 = 5 - 1
[1,2,3,5,4] => 3 = 4 - 1
[1,2,4,3,5] => 3 = 4 - 1
[1,2,4,5,3] => 2 = 3 - 1
[1,2,5,3,4] => 2 = 3 - 1
[1,2,5,4,3] => 2 = 3 - 1
[1,3,2,4,5] => 3 = 4 - 1
[1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => 2 = 3 - 1
[1,3,4,5,2] => 1 = 2 - 1
[1,3,5,2,4] => 1 = 2 - 1
[1,3,5,4,2] => 1 = 2 - 1
[1,4,2,3,5] => 2 = 3 - 1
[1,4,2,5,3] => 1 = 2 - 1
[1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => 1 = 2 - 1
[1,4,5,2,3] => 1 = 2 - 1
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000011
(load all 127 compositions to match this statistic)
(load all 127 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 2
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 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: St000287
(load all 170 compositions to match this statistic)
(load all 170 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000287: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000287: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,2] => ([],2)
=> 2
[2,1] => ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> 3
[1,3,2] => ([(1,2)],3)
=> 2
[2,1,3] => ([(1,2)],3)
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => ([],4)
=> 4
[1,2,4,3] => ([(2,3)],4)
=> 3
[1,3,2,4] => ([(2,3)],4)
=> 3
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> 3
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => ([],5)
=> 5
[1,2,3,5,4] => ([(3,4)],5)
=> 4
[1,2,4,3,5] => ([(3,4)],5)
=> 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
Description
The number of connected components of a graph.
Matching statistic: St000546
(load all 356 compositions to match this statistic)
(load all 356 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [2,1] => 1 = 2 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [3,2,1] => 2 = 3 - 1
[1,3,2] => [3,1,2] => 1 = 2 - 1
[2,1,3] => [2,3,1] => 1 = 2 - 1
[2,3,1] => [2,1,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,2,4,3] => [4,3,1,2] => 2 = 3 - 1
[1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
[1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[2,1,4,3] => [3,4,1,2] => 1 = 2 - 1
[2,3,1,4] => [3,2,4,1] => 1 = 2 - 1
[2,3,4,1] => [3,2,1,4] => 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => 0 = 1 - 1
[2,4,3,1] => [3,1,2,4] => 0 = 1 - 1
[3,1,2,4] => [2,4,3,1] => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => 0 = 1 - 1
[3,2,1,4] => [2,3,4,1] => 1 = 2 - 1
[3,2,4,1] => [2,3,1,4] => 0 = 1 - 1
[3,4,1,2] => [2,1,4,3] => 0 = 1 - 1
[3,4,2,1] => [2,1,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => 0 = 1 - 1
[4,2,1,3] => [1,3,4,2] => 0 = 1 - 1
[4,2,3,1] => [1,3,2,4] => 0 = 1 - 1
[4,3,1,2] => [1,2,4,3] => 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [5,4,3,2,1] => 4 = 5 - 1
[1,2,3,5,4] => [5,4,3,1,2] => 3 = 4 - 1
[1,2,4,3,5] => [5,4,2,3,1] => 3 = 4 - 1
[1,2,4,5,3] => [5,4,2,1,3] => 2 = 3 - 1
[1,2,5,3,4] => [5,4,1,3,2] => 2 = 3 - 1
[1,2,5,4,3] => [5,4,1,2,3] => 2 = 3 - 1
[1,3,2,4,5] => [5,3,4,2,1] => 3 = 4 - 1
[1,3,2,5,4] => [5,3,4,1,2] => 2 = 3 - 1
[1,3,4,2,5] => [5,3,2,4,1] => 2 = 3 - 1
[1,3,4,5,2] => [5,3,2,1,4] => 1 = 2 - 1
[1,3,5,2,4] => [5,3,1,4,2] => 1 = 2 - 1
[1,3,5,4,2] => [5,3,1,2,4] => 1 = 2 - 1
[1,4,2,3,5] => [5,2,4,3,1] => 2 = 3 - 1
[1,4,2,5,3] => [5,2,4,1,3] => 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => 2 = 3 - 1
[1,4,3,5,2] => [5,2,3,1,4] => 1 = 2 - 1
[1,4,5,2,3] => [5,2,1,4,3] => 1 = 2 - 1
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St000007
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [2,1] => 2
[2,1] => [1,1,0,0]
=> [1,2] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: 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
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1
[1,2] => ([],2)
=> [1,1]
=> 2
[2,1] => ([(0,1)],2)
=> [2]
=> 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 3
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 4
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 3
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 3
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 3
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 5
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 4
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2
Description
The length of the partition.
Matching statistic: St000025
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000084
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [[]]
=> 1
[1,2] => [1,0,1,0]
=> [[],[]]
=> 2
[2,1] => [1,1,0,0]
=> [[[]]]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [[[],[]]]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 2
Description
The number of subtrees.
Matching statistic: St000286
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2,1] => ([(0,1)],2)
=> 2
[2,1] => [1,2] => ([],2)
=> 1
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> 1
[3,2,1] => [1,2,3] => ([],3)
=> 1
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> 1
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([],4)
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [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
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
Description
The number of connected components of the complement of a graph.
The complement of a graph is the graph on the same vertex set with complementary edges.
The following 120 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000383The last part of an integer composition. St000544The cop number of a graph. St000553The number of blocks of a graph. St000843The decomposition number of a perfect matching. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001363The Euler characteristic of a graph according to Knill. St001462The number of factors of a standard tableaux under concatenation. St000237The number of small exceedances. St000439The position of the first down step of a Dyck path. 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. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000172The Grundy number of a graph. St000239The number of small weak excedances. St000273The domination number of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000822The Hadwiger number of the graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St001029The size of the core of a graph. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001829The common independence number of a graph. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000672The number of minimal elements in Bruhat order not less than the permutation. St000675The number of centered multitunnels of a Dyck path. St000738The first entry in the last row of a standard tableau. St000883The number of longest increasing subsequences of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. 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. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001777The number of weak descents in an integer composition. St000717The number of ordinal summands of a poset. St000061The number of nodes on the left branch of a binary tree. St000504The cardinality of the first block of a set partition. St000654The first descent of a permutation. St000906The length of the shortest maximal chain in a poset. St000914The sum of the values of the Möbius function of a poset. St000990The first ascent of a permutation. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001812The biclique partition number of a graph. St001828The Euler characteristic of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001330The hat guessing number of a graph. St000648The number of 2-excedences 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000456The monochromatic index of a connected graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000924The number of topologically connected components of a perfect matching. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000732The number of double deficiencies of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000898The number of maximal entries in the last diagonal of the monotone triangle. St001570The minimal number of edges to add to make a graph Hamiltonian. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001889The size of the connectivity set of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function.
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