Your data matches 158 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000168
(load all 20 compositions to match this statistic)
Mapping
St000168: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => 0
[[],[]]
 => 0
[[[]]]
 => 1
[[],[],[]]
 => 0
[[],[[]]]
 => 1
[[[]],[]]
 => 1
[[[],[]]]
 => 1
[[[[]]]]
 => 2
[[],[],[],[]]
 => 0
[[],[],[[]]]
 => 1
[[],[[]],[]]
 => 1
[[],[[],[]]]
 => 1
[[],[[[]]]]
 => 2
[[[]],[],[]]
 => 1
[[[]],[[]]]
 => 2
[[[],[]],[]]
 => 1
[[[[]]],[]]
 => 2
[[[],[],[]]]
 => 1
[[[],[[]]]]
 => 2
[[[[]],[]]]
 => 2
[[[[],[]]]]
 => 2
[[[[[]]]]]
 => 3
[[],[],[],[],[]]
 => 0
[[],[],[],[[]]]
 => 1
[[],[],[[]],[]]
 => 1
[[],[],[[],[]]]
 => 1
[[],[],[[[]]]]
 => 2
[[],[[]],[],[]]
 => 1
[[],[[]],[[]]]
 => 2
[[],[[],[]],[]]
 => 1
[[],[[[]]],[]]
 => 2
[[],[[],[],[]]]
 => 1
[[],[[],[[]]]]
 => 2
[[],[[[]],[]]]
 => 2
[[],[[[],[]]]]
 => 2
[[],[[[[]]]]]
 => 3
[[[]],[],[],[]]
 => 1
[[[]],[],[[]]]
 => 2
[[[]],[[]],[]]
 => 2
[[[]],[[],[]]]
 => 2
[[[]],[[[]]]]
 => 3
[[[],[]],[],[]]
 => 1
[[[[]]],[],[]]
 => 2
[[[],[]],[[]]]
 => 2
[[[[]]],[[]]]
 => 3
[[[],[],[]],[]]
 => 1
[[[],[[]]],[]]
 => 2
[[[[]],[]],[]]
 => 2
[[[[],[]]],[]]
 => 2
[[[[[]]]],[]]
 => 3
Description
The number of internal nodes of an ordered tree. A node is internal if it is neither the root nor a leaf.
Matching statistic: St000024
(load all 82 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => 0
[[[]]]
 => [1,1,0,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 2
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 3
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000167
(load all 17 compositions to match this statistic)
Mapping
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
St000167: 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
[[],[[]]]
 => [[[],[]]]
 => 2 = 1 + 1
[[[]],[]]
 => [[],[[]]]
 => 2 = 1 + 1
[[[],[]]]
 => [[[]],[]]
 => 2 = 1 + 1
[[[[]]]]
 => [[],[],[]]
 => 3 = 2 + 1
[[],[],[],[]]
 => [[[[[]]]]]
 => 1 = 0 + 1
[[],[],[[]]]
 => [[[[],[]]]]
 => 2 = 1 + 1
[[],[[]],[]]
 => [[[],[[]]]]
 => 2 = 1 + 1
[[],[[],[]]]
 => [[[[]],[]]]
 => 2 = 1 + 1
[[],[[[]]]]
 => [[[],[],[]]]
 => 3 = 2 + 1
[[[]],[],[]]
 => [[],[[[]]]]
 => 2 = 1 + 1
[[[]],[[]]]
 => [[],[[],[]]]
 => 3 = 2 + 1
[[[],[]],[]]
 => [[[]],[[]]]
 => 2 = 1 + 1
[[[[]]],[]]
 => [[],[],[[]]]
 => 3 = 2 + 1
[[[],[],[]]]
 => [[[[]]],[]]
 => 2 = 1 + 1
[[[],[[]]]]
 => [[[],[]],[]]
 => 3 = 2 + 1
[[[[]],[]]]
 => [[],[[]],[]]
 => 3 = 2 + 1
[[[[],[]]]]
 => [[[]],[],[]]
 => 3 = 2 + 1
[[[[[]]]]]
 => [[],[],[],[]]
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [[[[[[]]]]]]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [[[[[],[]]]]]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [[[[],[[]]]]]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [[[[[]],[]]]]
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [[[[],[],[]]]]
 => 3 = 2 + 1
[[],[[]],[],[]]
 => [[[],[[[]]]]]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [[[],[[],[]]]]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [[[[]],[[]]]]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [[[],[],[[]]]]
 => 3 = 2 + 1
[[],[[],[],[]]]
 => [[[[[]]],[]]]
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [[[[],[]],[]]]
 => 3 = 2 + 1
[[],[[[]],[]]]
 => [[[],[[]],[]]]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [[[[]],[],[]]]
 => 3 = 2 + 1
[[],[[[[]]]]]
 => [[[],[],[],[]]]
 => 4 = 3 + 1
[[[]],[],[],[]]
 => [[],[[[[]]]]]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [[],[[[],[]]]]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [[],[[],[[]]]]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [[],[[[]],[]]]
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [[],[[],[],[]]]
 => 4 = 3 + 1
[[[],[]],[],[]]
 => [[[]],[[[]]]]
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => 3 = 2 + 1
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => 3 = 2 + 1
[[[[]]],[[]]]
 => [[],[],[[],[]]]
 => 4 = 3 + 1
[[[],[],[]],[]]
 => [[[[]]],[[]]]
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [[[],[]],[[]]]
 => 3 = 2 + 1
[[[[]],[]],[]]
 => [[],[[]],[[]]]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [[[]],[],[[]]]
 => 3 = 2 + 1
[[[[[]]]],[]]
 => [[],[],[],[[]]]
 => 4 = 3 + 1
Description
The number of leaves of an ordered tree. This is the number of nodes which do not have any children.
Matching statistic: St000443
(load all 82 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => 2 = 1 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
Description
The number of long tunnels of a Dyck path. A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Matching statistic: St001007
(load all 82 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => 2 = 1 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001187
(load all 82 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001187: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => 2 = 1 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
Description
The number of simple modules with grade at least one in the corresponding Nakayama algebra.
Matching statistic: St001224
(load all 82 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001224: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => 2 = 1 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
Description
Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the first Ext-group between X and the regular module.
Matching statistic: St000021
(load all 41 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => 0
[[],[]]
 => [[.,.],.]
 => [1,2] => 0
[[[]]]
 => [.,[.,.]]
 => [2,1] => 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => 0
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 1
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 2
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 2
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 2
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 3
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => 0
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => 2
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => 2
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => 2
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => 2
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => 2
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => 3
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => 2
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => 2
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => 3
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 2
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 3
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 2
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 2
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 3
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000053
(load all 81 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
Description
The number of valleys of the Dyck path.
Matching statistic: St000120
(load all 8 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 2
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
Description
The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
The following 148 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000632The jump number of the poset. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001622The number of join-irreducible elements of a lattice. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000080The rank of the poset. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000214The number of adjacencies of a permutation. St000292The number of ascents of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000386The number of factors DDU in a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000662The staircase size of the code of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001427The number of descents of a signed permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000007The number of saliances of the permutation. St000010The length of the partition. St000031The number of cycles in the cycle decomposition of a permutation. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000164The number of short pairs. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000377The dinv defect of an integer partition. St000390The number of runs of ones in a binary word. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St001029The size of the core of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001581The achromatic number of a graph. St001820The size of the image of the pop stack sorting operator. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St000702The number of weak deficiencies of a permutation. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000216The absolute length of a permutation. St000288The number of ones in a binary word. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001480The number of simple summands of the module J^2/J^3. St000061The number of nodes on the left branch of a binary tree. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000925The number of topologically connected components of a set partition. St001812The biclique partition number of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001875The number of simple modules with projective dimension at most 1. St000159The number of distinct parts of the integer partition. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. 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. St000746The number of pairs with odd minimum in a perfect matching. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000242The number of indices that are not cyclical small weak excedances. St001152The number of pairs with even minimum in a perfect matching. St000619The number of cyclic descents of a permutation. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001896The number of right descents of a signed permutations. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001720The minimal length of a chain of small intervals in a lattice.