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Your data matches 234 different statistics following compositions of up to 3 maps.
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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
[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 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 - 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: 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 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 2 = 3 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 0 = 1 - 1
[1,4,2,3] => 0 = 1 - 1
[1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => 0 = 1 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 0 = 1 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 0 = 1 - 1
[3,1,4,2] => 0 = 1 - 1
[3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 1 = 2 - 1
[3,4,2,1] => 2 = 3 - 1
[4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 1 = 2 - 1
[4,2,3,1] => 2 = 3 - 1
[4,3,1,2] => 2 = 3 - 1
[4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 0 = 1 - 1
[1,2,5,3,4] => 0 = 1 - 1
[1,2,5,4,3] => 0 = 1 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => 0 = 1 - 1
[1,3,4,5,2] => 0 = 1 - 1
[1,3,5,2,4] => 0 = 1 - 1
[1,3,5,4,2] => 0 = 1 - 1
[1,4,2,3,5] => 0 = 1 - 1
[1,4,2,5,3] => 0 = 1 - 1
[1,4,3,2,5] => 0 = 1 - 1
[1,4,3,5,2] => 0 = 1 - 1
[1,4,5,2,3] => 0 = 1 - 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: 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
[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: 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
[1,2] => ([],2)
=> 1
[2,1] => ([(0,1)],2)
=> 2
[1,2,3] => ([],3)
=> 1
[1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => ([],4)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[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)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => ([],5)
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 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
[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: 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
[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 4 compositions to match this statistic)
(load all 4 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 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
[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 5 compositions to match this statistic)
(load all 5 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.
The following 224 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. St001461The number of topologically connected components of the chord diagram of a permutation. 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. 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. 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. 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. 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. 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. 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. 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. St000925The number of topologically connected components of a set partition. 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. 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. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001933The largest multiplicity of a part in an integer partition. St000026The position of the first return of a Dyck path. St000160The multiplicity of the smallest part of a partition. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000627The exponent of a binary word. St000667The greatest common divisor of the parts of the partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001267The length of the Lyndon factorization of the binary word. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001571The Cartan determinant of the integer partition. St001884The number of borders of a binary word. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001568The smallest positive integer that does not appear twice in the partition. St000648The number of 2-excedences of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000993The multiplicity of the largest part of an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. 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. St000455The second largest eigenvalue of a graph if it is integral. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001657The number of twos in an integer partition. St000284The Plancherel distribution on integer partitions. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a 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. 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000707The product of the factorials of the parts. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000618The number of self-evacuating tableaux of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001877Number of indecomposable injective modules with projective dimension 2. St001004The number of indices that are either left-to-right maxima or right-to-left minima. 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. 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. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St001875The number of simple modules with projective dimension at most 1. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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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