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Your data matches 284 different statistics following compositions of up to 3 maps.
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Matching statistic: St000234
(load all 106 compositions to match this statistic)
(load all 106 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 0
[[.,.],.]
=> [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => 0
[[.,.],[.,.]]
=> [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 3
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 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 1
[[.,.],.]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => 2
[.,[[.,.],.]]
=> [2,3,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 3
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 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: St001185
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
Matching statistic: St001640
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 0
[[.,.],.]
=> [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => 0
[[.,.],[.,.]]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000007
(load all 142 compositions to match this statistic)
(load all 142 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => 2 = 1 + 1
[[.,.],.]
=> [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => 3 = 2 + 1
[.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3 = 2 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3 = 2 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 4 = 3 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 4 = 3 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4 = 3 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 4 = 3 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 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: St000011
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding 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
[.,.]
=> [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[.,.],.]
=> [1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 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: St000025
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,1,0,0]
=> 2 = 1 + 1
[[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
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: St000054
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => 2 = 1 + 1
[[.,.],.]
=> [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => 3 = 2 + 1
[.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[[.,.],[.,.]]
=> [1,3,2] => 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 4 = 3 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1 = 0 + 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000056
(load all 106 compositions to match this statistic)
(load all 106 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => 1 = 0 + 1
[[.,.],.]
=> [1,2] => 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => 1 = 0 + 1
[[.,.],[.,.]]
=> [1,3,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 3 = 2 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 4 = 3 + 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: St000084
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [[]]
=> 1 = 0 + 1
[.,[.,.]]
=> [[[]]]
=> 1 = 0 + 1
[[.,.],.]
=> [[],[]]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [[[],[]]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[],[[]]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [[[]],[]]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [[],[],[]]
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 3 = 2 + 1
[[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [[],[[[],[]]]]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [[],[[],[[]]]]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [[],[[[]],[]]]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[]],[[],[]]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[[]]],[[]]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[]],[[]]]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
[[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 4 = 3 + 1
Description
The number of subtrees.
The following 274 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. 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. 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. St001461The number of topologically connected components of the chord diagram of a permutation. 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. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000133The "bounce" of a permutation. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001479The number of bridges of a graph. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a 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. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000297The number of leading ones in a binary word. St000382The first part of an integer composition. St000383The last part of an integer composition. St000544The cop number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000759The smallest missing part in an integer partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000203The number of external nodes of a binary tree. St000675The number of centered multitunnels of a Dyck path. St000738The first entry in the last row of a standard tableau. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000245The number of ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000445The number of rises of length 1 of a Dyck path. St000446The disorder of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000783The side length of the largest staircase partition fitting into a partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001358The largest degree of a regular subgraph of a graph. St001484The number of singletons of an integer partition. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001777The number of weak descents in an integer composition. St000013The height of a Dyck path. St000026The position of the first return of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000352The Elizalde-Pak rank of a permutation. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000822The Hadwiger number of the graph. St000883The number of longest increasing subsequences of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama 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:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001963The tree-depth of a graph. St000504The cardinality of the first block of a set partition. St000734The last entry in the first row of a standard tableau. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000061The number of nodes on the left branch of a binary tree. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000990The first ascent of a permutation. St000717The number of ordinal summands of a poset. 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. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000653The last descent of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St001480The number of simple summands of the module J^2/J^3. St000702The number of weak deficiencies of a permutation. St000993The multiplicity of the largest part of an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000159The number of distinct parts of the integer partition. St000475The number of parts equal to 1 in a partition. St001812The biclique partition number of a graph. St000674The number of hills of a Dyck path. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001330The hat guessing number of a graph. St000703The number of deficiencies of a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001280The number of parts of an integer partition that are at least two. St001432The order dimension of the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000648The number of 2-excedences of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000454The largest eigenvalue of a graph if it is integral. St000117The number of centered tunnels of a Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a 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. St000260The radius of a connected graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000941The number of characters of the symmetric group whose value on the partition is even. St001651The Frankl number of a lattice. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000924The number of topologically connected components of a perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000022The number of fixed points of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000241The number of cyclical small excedances. St000338The number of pixed points of a permutation. St000456The monochromatic index of a connected graph. St001552The number of inversions between excedances and fixed points of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last 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. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001498The normalised height of a Nakayama algebra with magnitude 1. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000259The diameter of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St001889The size of the connectivity set of a signed permutation. St001862The number of crossings of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001060The distinguishing index of a graph. 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. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001904The length of the initial strictly increasing segment of a parking function. St001557The number of inversions of the second entry of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001937The size of the center of a parking function. 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. St001621The number of atoms of a lattice.
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