Your data matches 198 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000974
(load all 10 compositions to match this statistic)
Mapping
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000974: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[]]
 => 1 = 2 - 1
[1,0,1,0]
 => [[],[]]
 => 0 = 1 - 1
[1,1,0,0]
 => [[[]]]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [[],[],[]]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [[],[[]]]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [[[]],[]]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [[[],[]]]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [[[[]]]]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => 1 = 2 - 1
Description
The length of the trunk of an ordered tree. This is the length of the path from the root to the first vertex which has not exactly one child.
Matching statistic: St000382
(load all 7 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2] => [2] => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => [1,2] => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => [3] => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,2] => 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [2,2] => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [1,3] => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [1,3] => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [4] => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,1,2] => 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,1,2] => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,2,2] => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,2,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2] => 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [1,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [2,3] => 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [1,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [1,1,3] => 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [2,3] => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [1,4] => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [1,4] => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [1,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [5] => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => [1,1,1,1,2] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => [2,1,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [1,2,1,2] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [1,2,1,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => [3,1,2] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [1,1,2,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [2,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [1,1,2,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [1,1,2,2] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [2,2,2] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => [1,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => [1,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => [1,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => [4,2] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => [2,1,3] => 2
[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] => [1,2,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => [1,2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => [3,3] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => [2,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => [1,1,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => [1,1,1,3] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => [2,1,3] => 2
[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] => [1,2,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => [1,2,3] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => [1,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => [3,3] => 3
Description
The first part of an integer composition.
Matching statistic: St000678
(load all 4 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000678: 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]
 => [1,0,1,0]
 => 2
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,0,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [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]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [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]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000297
(load all 14 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 1 => 1 => 1 = 2 - 1
[1,0,1,0]
 => [1,1] => 11 => 11 => 2 = 3 - 1
[1,1,0,0]
 => [2] => 10 => 00 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1] => 111 => 111 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,2] => 110 => 001 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1] => 101 => 100 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3] => 100 => 010 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3] => 100 => 010 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 1111 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 0011 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 1001 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,3] => 1100 => 0101 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 0101 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 1100 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 0000 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [3,1] => 1001 => 1010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [4] => 1000 => 0110 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [4] => 1000 => 0110 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 1010 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [4] => 1000 => 0110 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [4] => 1000 => 0110 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4] => 1000 => 0110 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 11111 => 11111 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 11110 => 00111 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 11101 => 10011 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 11100 => 01011 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 11100 => 01011 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 11011 => 11001 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 11010 => 00001 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 11001 => 10101 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 11000 => 01101 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 11000 => 01101 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 11001 => 10101 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 11000 => 01101 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 11000 => 01101 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 01101 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 11100 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 00100 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 10101 => 10000 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 10100 => 01000 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 10100 => 01000 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 10011 => 11010 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => 10010 => 00010 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => 10001 => 10110 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => 10000 => 01110 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => 10000 => 01110 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 10001 => 10110 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => 10000 => 01110 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => 10000 => 01110 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => 10000 => 01110 => 0 = 1 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000445
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000475
(load all 5 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => [1]
 => 1 = 2 - 1
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => [2]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => ([],2)
 => [1,1]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => [2,1]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => [1,1,1]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => [1,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 2 - 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001107
(load all 24 compositions to match this statistic)
Mapping
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ? = 2 - 1
[1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => 0 = 1 - 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000674
(load all 28 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => ? = 2 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
Description
The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000383
(load all 11 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 75% ●values known / values provided: 99%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => []
 => []
 => [] => ? = 2
[1,0,1,0]
 => [1]
 => [1,0,1,0]
 => [1,1] => 1
[1,1,0,0]
 => []
 => []
 => [] => ? = 3
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1] => 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1] => 1
[1,1,1,0,0,0]
 => []
 => []
 => [] => ? = 4
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1] => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1] => 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [] => ? = 5
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 2
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [] => ? = 6
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => [] => ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => [] => ? = 8
Description
The last part of an integer composition.
Matching statistic: St000823
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000823: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 99%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => []
 => []
 => {}
 => ? = 2
[1,0,1,0]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 1
[1,1,0,0]
 => []
 => []
 => {}
 => ? = 3
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 1
[1,1,1,0,0,0]
 => []
 => []
 => {}
 => ? = 4
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 3
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => {}
 => ? = 5
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 3
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => {}
 => ? = 6
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => {}
 => ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => {}
 => ? = 8
Description
The number of unsplittable factors of the set partition. Let $\pi$ be a set partition of $m$ into $k$ parts and $\sigma$ a set partition of $n$ into $\ell$ parts. Then their split product is the set partition of $m+n$ with blocks $$ B_1 \cup (C_1+m),\dots,B_k \cup (C_k+m), C_{k+1}+m,\dots,C_\ell+m $$ if $k\leq \ell$ and $$ B_1 \cup (C_1+m),\dots,B_\ell \cup (C_\ell+m), B_{\ell+1},\dots,B_k $$ otherwise.
The following 188 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001691The number of kings in a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000993The multiplicity of the largest part of an integer partition. St000022The number of fixed points of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000288The number of ones in a binary word. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000011The number of touch points (or returns) of a Dyck path. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000990The first ascent of a permutation. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001571The Cartan determinant of the integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001933The largest multiplicity of a part in an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000617The number of global maxima of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000700The protection number of an ordered tree. St000706The product of the factorials of the multiplicities of an integer partition. St000053The number of valleys of the Dyck path. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001732The number of peaks visible from the left. St001884The number of borders of a binary word. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001924The number of cells in an integer partition whose arm and leg length coincide. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St000654The first descent of a permutation. St001128The exponens consonantiae of a partition. St000873The aix statistic of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000895The number of ones on the main diagonal of an alternating sign matrix. St001733The number of weak left to right maxima of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000100The number of linear extensions of a poset. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001280The number of parts of an integer partition that are at least two. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001890The maximum magnitude of the Möbius function of a poset. St000657The smallest part of an integer composition. St000765The number of weak records in an integer composition. St001657The number of twos in an integer partition. St000729The minimal arc length of a set partition. St000221The number of strong fixed points of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000234The number of global ascents of a permutation. St000315The number of isolated vertices of a graph. St000461The rix statistic of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000273The domination number of a graph. St000456The monochromatic index of a connected graph. St000544The cop number of a graph. St000916The packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001829The common independence number of a graph. St000117The number of centered tunnels of a Dyck path. St000241The number of cyclical small excedances. St000338The number of pixed points of a permutation. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000363The number of minimal vertex covers of a graph. St001672The restrained domination number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001479The number of bridges of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000056The decomposition (or block) number of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000989The number of final rises of a permutation. St000546The number of global descents of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima 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. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000054The first entry of the permutation. St001075The minimal size of a block of a set partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000314The number of left-to-right-maxima of a permutation. St000717The number of ordinal summands of a poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000740The last entry of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001118The acyclic chromatic index of a graph. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001889The size of the connectivity set of a signed permutation. St001060The distinguishing index of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001948The number of augmented double ascents of a permutation.