Identifier
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Mp00199:
Dyck paths
—prime Dyck path⟶
Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000234: Permutations ⟶ ℤ (values match St000056The decomposition (or block) number of a permutation.)
Values
[1,0] => [1,1,0,0] => [1,2] => 1
[1,0,1,0] => [1,1,0,1,0,0] => [3,1,2] => 0
[1,1,0,0] => [1,1,1,0,0,0] => [1,2,3] => 2
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => 0
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => 1
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => 1
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => 0
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => 0
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => 1
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => 0
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => 0
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 2
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => 1
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => 2
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => 0
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => 0
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => 1
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => 2
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => 1
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => 0
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 4
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,1,2] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [3,4,5,1,2,6] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [3,4,1,6,2,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [3,4,6,1,2,5] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,1,2,5,6] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [3,1,5,6,2,4] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4,6] => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [3,5,1,6,2,4] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [3,5,6,1,2,4] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4,6] => 1
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [3,1,2,6,4,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,6,2,4,5] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [3,6,1,2,4,5] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [3,1,2,4,5,6] => 3
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,5,6,2,3] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => 2
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,6,3,5] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,6,2,3,5] => 1
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,2,3,5,6] => 3
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,1,5,6,2,3] => 0
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [4,1,5,2,3,6] => 1
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,5,1,6,2,3] => 0
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => 0
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,1,2,3,6] => 1
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,2,6,3,5] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [4,1,2,3,5,6] => 2
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,2,5,3,4,6] => 3
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,6,3,4] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,5,6,2,3,4] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,5,2,3,4,6] => 2
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,1,2,6,3,4] => 0
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [5,1,6,2,3,4] => 0
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [5,1,2,3,4,6] => 1
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => 3
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,2,6,3,4,5] => 2
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,1,2] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [3,4,5,6,1,2,7] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [3,4,5,1,7,2,6] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [3,4,5,7,1,2,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,1,2,6,7] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [3,4,1,6,7,2,5] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [3,4,1,6,2,5,7] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [3,4,6,1,7,2,5] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [3,4,6,7,1,2,5] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5,7] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [3,4,1,2,7,5,6] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [3,4,1,7,2,5,6] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [3,4,7,1,2,5,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [3,4,1,2,5,6,7] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [3,1,5,6,7,2,4] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [3,1,5,6,2,4,7] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [3,1,5,2,7,4,6] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [3,1,5,7,2,4,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [3,1,5,2,4,6,7] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [3,5,1,6,7,2,4] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [3,5,1,6,2,4,7] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [3,5,6,1,7,2,4] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [3,5,6,7,1,2,4] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [3,5,6,1,2,4,7] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,7,4,6] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [3,5,1,7,2,4,6] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [3,5,7,1,2,4,6] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [3,5,1,2,4,6,7] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [3,1,2,6,7,4,5] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [3,1,2,6,4,5,7] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [3,1,6,2,7,4,5] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [3,1,6,7,2,4,5] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [3,1,6,2,4,5,7] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [3,6,1,2,7,4,5] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [3,6,1,7,2,4,5] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [3,6,7,1,2,4,5] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [3,6,1,2,4,5,7] => 1
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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.
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.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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