Your data matches 68 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000329
(load all 15 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000329: 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]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => 1
[[],[],[],[]]
 => [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]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[]],[],[]]
 => [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]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 3
[[[],[[]]]]
 => [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]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 2
[[],[],[],[],[]]
 => [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]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[[],[[]],[],[]]
 => [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]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[[],[[],[[]]]]
 => [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]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[[[]],[],[],[]]
 => [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]
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[[],[[]]],[]]
 => [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]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000053
(load all 7 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ 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]
 => [1,1,0,0,1,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[[],[],[],[]]
 => [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]
 => [1,1,1,0,0,0,1,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 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]
 => [1,1,1,0,1,0,0,0]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[],[],[],[],[]]
 => [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]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
Description
The number of valleys of the Dyck path.
Matching statistic: St000155
(load all 3 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => 0
[[],[]]
 => [1,0,1,0]
 => [1,2] => 0
[[[]]]
 => [1,1,0,0]
 => [2,1] => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 2
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St001169
(load all 7 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001169: 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]
 => [1,1,0,0,1,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[[],[],[],[]]
 => [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]
 => [1,1,1,0,0,0,1,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 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]
 => [1,1,1,0,1,0,0,0]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[],[],[],[],[]]
 => [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]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000015
(load all 7 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
Description
The number of peaks of a Dyck path.
Matching statistic: St000676
(load all 8 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 4 = 3 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 4 = 3 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 3 = 2 + 1
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001068
(load all 7 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000021
(load all 2 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —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]
 => [1,0]
 => [1] => 0
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,2] => 0
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [2,1] => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 3
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 3
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 2
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: St000024
(load all 9 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 3
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 2
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: St000052
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => [1,1,0,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 2
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001153The number of blocks with even minimum in a set partition. 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. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered 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. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to 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. St000542The number of left-to-right-minima of a permutation. St000912The number of maximal antichains in a poset. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. 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. St000925The number of topologically connected components of a set partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000159The number of distinct parts of the integer partition. St000374The number of exclusive right-to-left minima of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St001152The number of pairs with even minimum in a perfect matching. St000991The number of right-to-left minima of a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001712The number of natural descents of a standard Young tableau.