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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000084
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ 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] => [[]]
=> 1 = 0 + 1
[1,1] => [[],[]]
=> 2 = 1 + 1
[1,2] => [[[]]]
=> 1 = 0 + 1
[2,1] => [[[]]]
=> 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> 1 = 0 + 1
Description
The number of subtrees.
Matching statistic: St000011
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [[],[]]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2] => [[[]]]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [[[]]]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 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: St000234
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [1,0]
=> [1] => 0
[1,1] => [[],[]]
=> [1,0,1,0]
=> [1,2] => 1
[1,2] => [[[]]]
=> [1,1,0,0]
=> [2,1] => 0
[2,1] => [[[]]]
=> [1,1,0,0]
=> [2,1] => 0
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 0
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 0
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 0
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
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: St000237
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00290: Parking functions —to ordered set partition⟶ Ordered set partitions
Mp00285: Ordered set partitions —to set partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00285: Ordered set partitions —to set partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [{1}] => {{1}}
=> [1] => 0
[1,1] => [{1,2}] => {{1,2}}
=> [2,1] => 1
[1,2] => [{1},{2}] => {{1},{2}}
=> [1,2] => 0
[2,1] => [{2},{1}] => {{1},{2}}
=> [1,2] => 0
[1,1,1] => [{1,2,3}] => {{1,2,3}}
=> [2,3,1] => 2
[1,1,2] => [{1,2},{3}] => {{1,2},{3}}
=> [2,1,3] => 1
[1,2,1] => [{1,3},{2}] => {{1,3},{2}}
=> [3,2,1] => 0
[2,1,1] => [{2,3},{1}] => {{1},{2,3}}
=> [1,3,2] => 1
[1,1,3] => [{1,2},{3}] => {{1,2},{3}}
=> [2,1,3] => 1
[1,3,1] => [{1,3},{2}] => {{1,3},{2}}
=> [3,2,1] => 0
[3,1,1] => [{2,3},{1}] => {{1},{2,3}}
=> [1,3,2] => 1
[1,2,2] => [{1},{2,3}] => {{1},{2,3}}
=> [1,3,2] => 1
[2,1,2] => [{2},{1,3}] => {{1,3},{2}}
=> [3,2,1] => 0
[2,2,1] => [{3},{1,2}] => {{1,2},{3}}
=> [2,1,3] => 1
[1,2,3] => [{1},{2},{3}] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => [{1},{3},{2}] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,1,3] => [{2},{1},{3}] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,3,1] => [{3},{1},{2}] => {{1},{2},{3}}
=> [1,2,3] => 0
[3,1,2] => [{2},{3},{1}] => {{1},{2},{3}}
=> [1,2,3] => 0
[3,2,1] => [{3},{2},{1}] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,1,1,1] => [{1,2,3,4}] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[1,1,1,2] => [{1,2,3},{4}] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[1,1,2,1] => [{1,2,4},{3}] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[1,2,1,1] => [{1,3,4},{2}] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[2,1,1,1] => [{2,3,4},{1}] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,1,1,3] => [{1,2,3},{4}] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[1,1,3,1] => [{1,2,4},{3}] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[1,3,1,1] => [{1,3,4},{2}] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[3,1,1,1] => [{2,3,4},{1}] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,1,1,4] => [{1,2,3},{4}] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[1,1,4,1] => [{1,2,4},{3}] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[1,4,1,1] => [{1,3,4},{2}] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[4,1,1,1] => [{2,3,4},{1}] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,1,2,2] => [{1,2},{3,4}] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[1,2,1,2] => [{1,3},{2,4}] => {{1,3},{2,4}}
=> [3,4,1,2] => 0
[1,2,2,1] => [{1,4},{2,3}] => {{1,4},{2,3}}
=> [4,3,2,1] => 1
[2,1,1,2] => [{2,3},{1,4}] => {{1,4},{2,3}}
=> [4,3,2,1] => 1
[2,1,2,1] => [{2,4},{1,3}] => {{1,3},{2,4}}
=> [3,4,1,2] => 0
[2,2,1,1] => [{3,4},{1,2}] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[1,1,2,3] => [{1,2},{3},{4}] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[1,1,3,2] => [{1,2},{4},{3}] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[1,2,1,3] => [{1,3},{2},{4}] => {{1,3},{2},{4}}
=> [3,2,1,4] => 0
[1,2,3,1] => [{1,4},{2},{3}] => {{1,4},{2},{3}}
=> [4,2,3,1] => 0
[1,3,1,2] => [{1,3},{4},{2}] => {{1,3},{2},{4}}
=> [3,2,1,4] => 0
[1,3,2,1] => [{1,4},{3},{2}] => {{1,4},{2},{3}}
=> [4,2,3,1] => 0
[2,1,1,3] => [{2,3},{1},{4}] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[2,1,3,1] => [{2,4},{1},{3}] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[2,3,1,1] => [{3,4},{1},{2}] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[3,1,1,2] => [{2,3},{4},{1}] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[3,1,2,1] => [{2,4},{3},{1}] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000546
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
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%
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
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%
Values
[1] => [[]]
=> [.,.]
=> [1] => 0
[1,1] => [[],[]]
=> [.,[.,.]]
=> [2,1] => 1
[1,2] => [[[]]]
=> [[.,.],.]
=> [1,2] => 0
[2,1] => [[[]]]
=> [[.,.],.]
=> [1,2] => 0
[1,1,1] => [[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 2
[1,1,2] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,2,1] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[2,1,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,1,3] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,3,1] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[3,1,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,2,2] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[2,2,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,2,3] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,3,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[2,1,3] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[2,3,1] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[3,1,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[3,2,1] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1,1,1] => [[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3
[1,1,1,2] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,2,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,2,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,1,3] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,3,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[3,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,1,4] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,4,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[4,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,2,2] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,2,1,2] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[1,2,2,1] => [[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[2,1,1,2] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,2,1] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[2,2,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,1,2,3] => [[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[1,1,3,2] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,2,1,3] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,3,1] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[1,3,1,2] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,3,2,1] => [[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[2,1,1,3] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,3,1] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[2,3,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[3,1,1,2] => [[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[3,1,2,1] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St001640
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
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%
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
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%
Values
[1] => [[]]
=> [.,.]
=> [1] => 0
[1,1] => [[],[]]
=> [[.,.],.]
=> [1,2] => 1
[1,2] => [[[]]]
=> [.,[.,.]]
=> [2,1] => 0
[2,1] => [[[]]]
=> [.,[.,.]]
=> [2,1] => 0
[1,1,1] => [[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 2
[1,1,2] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,2,1] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[2,1,1] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,3] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,3,1] => [[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[3,1,1] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,2,2] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[2,1,2] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[2,2,1] => [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,2,3] => [[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[1,3,2] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[2,1,3] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[2,3,1] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[3,1,2] => [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[3,2,1] => [[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[1,1,1,1] => [[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 3
[1,1,1,2] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,2,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,2,1,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[2,1,1,1] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,1,3] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,3,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,3,1,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[3,1,1,1] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,1,4] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,4,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,4,1,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[4,1,1,1] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,2,2] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,2,1,2] => [[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[1,2,2,1] => [[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[2,1,1,2] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[2,1,2,1] => [[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[2,2,1,1] => [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,2,3] => [[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[1,1,3,2] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,2,1,3] => [[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,2,3,1] => [[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[1,3,1,2] => [[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,3,2,1] => [[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 0
[2,1,1,3] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[2,1,3,1] => [[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[2,3,1,1] => [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[3,1,1,2] => [[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[3,1,2,1] => [[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
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%
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
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%
Values
[1] => [[]]
=> [.,.]
=> [1] => 1 = 0 + 1
[1,1] => [[],[]]
=> [.,[.,.]]
=> [2,1] => 2 = 1 + 1
[1,2] => [[[]]]
=> [[.,.],.]
=> [1,2] => 1 = 0 + 1
[2,1] => [[[]]]
=> [[.,.],.]
=> [1,2] => 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 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: St000025
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —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%
Mp00051: Ordered trees —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%
Values
[1] => [[]]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[1,2] => [[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 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: St000056
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [1,0]
=> [1] => 1 = 0 + 1
[1,1] => [[],[]]
=> [1,0,1,0]
=> [1,2] => 2 = 1 + 1
[1,2] => [[[]]]
=> [1,1,0,0]
=> [2,1] => 1 = 0 + 1
[2,1] => [[[]]]
=> [1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 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: St000843
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000843: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000843: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> [1,0]
=> [(1,2)]
=> 1 = 0 + 1
[1,1] => [[],[]]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2 = 1 + 1
[1,2] => [[[]]]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 0 + 1
[2,1] => [[[]]]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1 = 0 + 1
Description
The decomposition number of a perfect matching.
This is the number of integers $i$ such that all elements in $\{1,\dots,i\}$ are matched among themselves.
Visually, it is the number of components of the arc diagram of the matching, where a component is a matching of a set of consecutive numbers $\{a,a+1,\dots,b\}$ such that there is no arc matching a number smaller than $a$ with a number larger than $b$.
E.g., $\{(1,6),(2,4),(3,5)\}$ is a hairpin under a single edge - crossing nested by a single arc. Thus, this matching has one component. However, $\{(1,2),(3,6),(4,5)\}$ is a single edge to the left of a ladder (a pair of nested edges), so it has two components.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. St000502The number of successions of a set partitions. St000061The number of nodes on the left branch of a binary tree. St000989The number of final rises of a permutation. St000504The cardinality of the first block of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St001947The number of ties in a parking function. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
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