Your data matches 79 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000141
(load all 19 compositions to match this statistic)
Mapping
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 2 = 3 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 2 = 3 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 2 = 3 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 2 = 3 - 1
[3,2,1,4] => 2 = 3 - 1
[3,2,4,1] => 2 = 3 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 2 = 3 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 3 = 4 - 1
[4,2,1,3] => 3 = 4 - 1
[4,2,3,1] => 3 = 4 - 1
[4,3,1,2] => 3 = 4 - 1
[4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => 1 = 2 - 1
[1,2,5,3,4] => 2 = 3 - 1
[1,2,5,4,3] => 2 = 3 - 1
[1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => 1 = 2 - 1
[1,3,4,2,5] => 1 = 2 - 1
[1,3,4,5,2] => 1 = 2 - 1
[1,3,5,2,4] => 2 = 3 - 1
[1,3,5,4,2] => 2 = 3 - 1
[1,4,2,3,5] => 2 = 3 - 1
[1,4,2,5,3] => 2 = 3 - 1
[1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => 2 = 3 - 1
[1,4,5,2,3] => 2 = 3 - 1
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000013
(load all 14 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 1
[1,2] => [1,0,1,0]
 => 1
[2,1] => [1,1,0,0]
 => 2
[1,2,3] => [1,0,1,0,1,0]
 => 1
[1,3,2] => [1,0,1,1,0,0]
 => 2
[2,1,3] => [1,1,0,0,1,0]
 => 2
[2,3,1] => [1,1,0,1,0,0]
 => 2
[3,1,2] => [1,1,1,0,0,0]
 => 3
[3,2,1] => [1,1,1,0,0,0]
 => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => 3
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000062
(load all 10 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => 1
[1,2] => [1,0,1,0]
 => [2,1] => 1
[2,1] => [1,1,0,0]
 => [1,2] => 2
[1,2,3] => [1,0,1,0,1,0]
 => [3,2,1] => 1
[1,3,2] => [1,0,1,1,0,0]
 => [2,3,1] => 2
[2,1,3] => [1,1,0,0,1,0]
 => [3,1,2] => 2
[2,3,1] => [1,1,0,1,0,0]
 => [2,1,3] => 2
[3,1,2] => [1,1,1,0,0,0]
 => [1,2,3] => 3
[3,2,1] => [1,1,1,0,0,0]
 => [1,2,3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 3
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000166
(load all 4 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [[]]
 => 1
[1,2] => [1,0,1,0]
 => [[],[]]
 => 1
[2,1] => [1,1,0,0]
 => [[[]]]
 => 2
[1,2,3] => [1,0,1,0,1,0]
 => [[],[],[]]
 => 1
[1,3,2] => [1,0,1,1,0,0]
 => [[],[[]]]
 => 2
[2,1,3] => [1,1,0,0,1,0]
 => [[[]],[]]
 => 2
[2,3,1] => [1,1,0,1,0,0]
 => [[[],[]]]
 => 2
[3,1,2] => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3
[3,2,1] => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => 3
Description
The depth minus 1 of an ordered tree. The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]].
Matching statistic: St000442
(load all 14 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => 1
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 3
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000527
(load all 4 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => ([],1)
 => 1
[1,2] => [1,0,1,0]
 => ([(0,1)],2)
 => 1
[2,1] => [1,1,0,0]
 => ([],2)
 => 2
[1,2,3] => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => 2
[2,1,3] => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => 2
[2,3,1] => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => 2
[3,1,2] => [1,1,1,0,0,0]
 => ([],3)
 => 3
[3,2,1] => [1,1,1,0,0,0]
 => ([],3)
 => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000720
(load all 5 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000720: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [(1,2)]
 => 1
[1,2] => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[2,1] => [1,1,0,0]
 => [(1,4),(2,3)]
 => 2
[1,2,3] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1
[1,3,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[2,1,3] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[2,3,1] => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 2
[3,1,2] => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[3,2,1] => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => 3
Description
The size of the largest partition in the oscillating tableau corresponding to the perfect matching. Equivalently, this is the maximal number of crosses in the corresponding triangular rook filling that can be covered by a rectangle.
Matching statistic: St001203
(load all 5 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => 1
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
Description
We associate to 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 Dyck path as follows: In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.
Matching statistic: St000094
(load all 4 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [[]]
 => 2 = 1 + 1
[1,2] => [1,0,1,0]
 => [[],[]]
 => 2 = 1 + 1
[2,1] => [1,1,0,0]
 => [[[]]]
 => 3 = 2 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [[],[],[]]
 => 2 = 1 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [[],[[]]]
 => 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [[[]],[]]
 => 3 = 2 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [[[],[]]]
 => 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 2 = 1 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 3 = 2 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 3 = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 4 = 3 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 4 = 3 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 3 = 2 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 3 = 2 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 3 = 2 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 3 = 2 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 4 = 3 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 4 = 3 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 4 = 3 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 4 = 3 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 4 = 3 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 4 = 3 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 5 = 4 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 5 = 4 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 5 = 4 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 5 = 4 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 5 = 4 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 5 = 4 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 2 = 1 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 3 = 2 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 3 = 2 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 4 = 3 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 4 = 3 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 3 = 2 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 4 = 3 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 4 = 3 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 4 = 3 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 4 = 3 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 4 = 3 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 4 = 3 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => 4 = 3 + 1
Description
The depth of an ordered tree.
Matching statistic: St000306
(load all 5 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => 0 = 1 - 1
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
Description
The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000662The staircase size of the code of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001494The Alon-Tarsi number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001717The largest size of an interval in a poset. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000651The maximal size of a rise in a permutation. St000845The maximal number of elements covered by an element in a poset. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St000730The maximal arc length of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001589The nesting number of a perfect matching. St001590The crossing number of a perfect matching. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000317The cycle descent number of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000983The length of the longest alternating subword. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.