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Your data matches 256 different statistics following compositions of up to 3 maps.
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Matching statistic: St000234
(load all 246 compositions to match this statistic)
(load all 246 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
[1,2] => 1
[2,1] => 0
[1,2,3] => 2
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 3
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 4
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 3
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 2
[1,4,3,5,2] => 1
[1,4,5,2,3] => 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: St000546
(load all 64 compositions to match this statistic)
(load all 64 compositions to match this statistic)
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
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: St000056
(load all 246 compositions to match this statistic)
(load all 246 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 = 0 + 1
[1,2] => 2 = 1 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 3 = 2 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 1 = 0 + 1
[3,1,2] => 1 = 0 + 1
[3,2,1] => 1 = 0 + 1
[1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => 3 = 2 + 1
[1,3,2,4] => 3 = 2 + 1
[1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => 3 = 2 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 2 = 1 + 1
[2,3,4,1] => 1 = 0 + 1
[2,4,1,3] => 1 = 0 + 1
[2,4,3,1] => 1 = 0 + 1
[3,1,2,4] => 2 = 1 + 1
[3,1,4,2] => 1 = 0 + 1
[3,2,1,4] => 2 = 1 + 1
[3,2,4,1] => 1 = 0 + 1
[3,4,1,2] => 1 = 0 + 1
[3,4,2,1] => 1 = 0 + 1
[4,1,2,3] => 1 = 0 + 1
[4,1,3,2] => 1 = 0 + 1
[4,2,1,3] => 1 = 0 + 1
[4,2,3,1] => 1 = 0 + 1
[4,3,1,2] => 1 = 0 + 1
[4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => 5 = 4 + 1
[1,2,3,5,4] => 4 = 3 + 1
[1,2,4,3,5] => 4 = 3 + 1
[1,2,4,5,3] => 3 = 2 + 1
[1,2,5,3,4] => 3 = 2 + 1
[1,2,5,4,3] => 3 = 2 + 1
[1,3,2,4,5] => 4 = 3 + 1
[1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => 3 = 2 + 1
[1,3,4,5,2] => 2 = 1 + 1
[1,3,5,2,4] => 2 = 1 + 1
[1,3,5,4,2] => 2 = 1 + 1
[1,4,2,3,5] => 3 = 2 + 1
[1,4,2,5,3] => 2 = 1 + 1
[1,4,3,2,5] => 3 = 2 + 1
[1,4,3,5,2] => 2 = 1 + 1
[1,4,5,2,3] => 2 = 1 + 1
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: St000011
(load all 34 compositions to match this statistic)
(load all 34 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 = 0 + 1
[1,2] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
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: St000286
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1 = 0 + 1
[1,2] => ([],2)
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
=> 1 = 0 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
=> 1 = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
=> 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
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.
Matching statistic: St000287
(load all 42 compositions to match this statistic)
(load all 42 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 = 0 + 1
[1,2] => ([],2)
=> 2 = 1 + 1
[2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => ([],3)
=> 3 = 2 + 1
[1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2,3,4] => ([],4)
=> 4 = 3 + 1
[1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,3,2,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => ([],5)
=> 5 = 4 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> 4 = 3 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> 4 = 3 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
Description
The number of connected components of a graph.
Matching statistic: St000237
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [2,1] => 1
[2,1] => [1,1,0,0]
=> [1,2] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => 2
[1,3,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => 0
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000007
(load all 23 compositions to match this statistic)
(load all 23 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 = 0 + 1
[1,2] => [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[2,1] => [1,1,0,0]
=> [1,2] => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3 = 2 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4 = 3 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3 = 2 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1 = 0 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 5 = 4 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 4 = 3 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4 = 3 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4 = 3 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 2 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2 = 1 + 1
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 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> 2 = 1 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> 1 = 0 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 3 = 2 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 5 = 4 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3 = 2 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000025
(load all 10 compositions to match this statistic)
(load all 10 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 = 0 + 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 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 = 4 + 1
[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 = 3 + 1
[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 = 3 + 1
[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 = 2 + 1
[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 = 2 + 1
[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 = 2 + 1
[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 = 3 + 1
[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 = 2 + 1
[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 = 2 + 1
[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 + 1
[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 + 1
[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 + 1
[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 = 2 + 1
[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 + 1
[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 = 2 + 1
[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 + 1
[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 = 1 + 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
The following 246 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000084The number of subtrees. 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. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. 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. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000214The number of adjacencies of a permutation. 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. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001358The largest degree of a regular subgraph of a graph. St001479The number of bridges 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. 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. 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. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000717The number of ordinal summands of a poset. St000061The number of nodes on the left branch of a binary tree. 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. St000932The number of occurrences of the pattern UDU in a Dyck path. St000989The number of final rises of a permutation. St001372The length of a longest cyclic run of ones of a binary word. 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. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001812The biclique partition number of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001828The Euler characteristic of a graph. St001330The hat guessing number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001525The number of symmetric hooks on the diagonal of a partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000376The bounce deficit of a Dyck path. St000691The number of changes of a binary word. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001091The number of parts in an integer partition whose next smaller part has the same size. St001139The number of occurrences of hills of size 2 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001252Half the sum of the even parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001524The degree of symmetry of a binary word. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000648The number of 2-excedences of a permutation. St000661The number of rises of length 3 of a Dyck path. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000934The 2-degree of an integer partition. St000369The dinv deficit of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001175The size of a partition minus the hook length of the base cell. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000699The toughness times the least common multiple of 1,. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the 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. St001541The Gini index of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001651The Frankl number of a lattice. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000681The Grundy value of Chomp on Ferrers diagrams. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000260The radius of a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000924The number of topologically connected components of a perfect matching. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000181The number of connected components of the Hasse diagram for the poset. St000732The number of double deficiencies of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. 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. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St000284The Plancherel distribution on integer partitions. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001561The value of the elementary symmetric function evaluated at 1. St000454The largest eigenvalue of a graph if it is integral. St001889The size of the connectivity set of a signed permutation. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001935The number of ascents in a parking function. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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