Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001185
(load all 3 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 0
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 0
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
Matching statistic: St000133
(load all 2 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => [1] => 0
[[],[]]
 => [1,0,1,0]
 => [2,1] => [1,2] => 1
[[[]]]
 => [1,1,0,0]
 => [1,2] => [2,1] => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1,3] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,3,1] => 0
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,3,2] => [3,1,2] => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [3,1,2] => [1,3,2] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [3,2,1,4] => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,4,1] => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [3,4,2,1] => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [4,2,1,3] => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [4,3,1,2] => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,4,3,1] => 0
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [4,1,3,2] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [1,4,3,2] => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,3,2,5,1] => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [4,3,5,1,2] => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [4,3,5,2,1] => 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [4,5,2,1,3] => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [4,5,2,3,1] => 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [4,2,1,5,3] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,2,5,3,1] => 0
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [4,5,1,3,2] => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [4,5,3,2,1] => 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [5,3,2,1,4] => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [5,3,2,4,1] => 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [5,3,4,1,2] => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [5,3,1,4,2] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [5,3,4,2,1] => 0
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [3,5,2,1,4] => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [5,4,2,1,3] => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [3,5,2,4,1] => 0
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [5,4,2,3,1] => 0
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,2,5,1,4] => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,5,4,1,2] => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [5,2,4,1,3] => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [2,5,4,1,3] => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [5,4,3,1,2] => 1
Description
The "bounce" of a permutation.
Matching statistic: St000054
Mapping
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [[]]
 => [1,0]
 => [1] => 1 = 0 + 1
[[],[]]
 => [[],[]]
 => [1,0,1,0]
 => [2,1] => 2 = 1 + 1
[[[]]]
 => [[[]]]
 => [1,1,0,0]
 => [1,2] => 1 = 0 + 1
[[],[],[]]
 => [[],[],[]]
 => [1,0,1,0,1,0]
 => [2,3,1] => 2 = 1 + 1
[[],[[]]]
 => [[[]],[]]
 => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 0 + 1
[[[]],[]]
 => [[],[[]]]
 => [1,0,1,1,0,0]
 => [2,1,3] => 2 = 1 + 1
[[[],[]]]
 => [[[],[]]]
 => [1,1,0,1,0,0]
 => [3,1,2] => 3 = 2 + 1
[[[[]]]]
 => [[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => 1 = 0 + 1
[[],[],[],[]]
 => [[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 2 = 1 + 1
[[],[],[[]]]
 => [[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1 = 0 + 1
[[],[[]],[]]
 => [[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[],[[],[]]]
 => [[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 3 = 2 + 1
[[],[[[]]]]
 => [[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 0 + 1
[[[]],[],[]]
 => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 1 + 1
[[[]],[[]]]
 => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 0 + 1
[[[],[]],[]]
 => [[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 2 = 1 + 1
[[[[]]],[]]
 => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 2 = 1 + 1
[[[],[],[]]]
 => [[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 3 = 2 + 1
[[[],[[]]]]
 => [[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1 = 0 + 1
[[[[]],[]]]
 => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 3 = 2 + 1
[[[[],[]]]]
 => [[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 4 = 3 + 1
[[[[[]]]]]
 => [[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 1 = 0 + 1
[[],[],[],[],[]]
 => [[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 2 = 1 + 1
[[],[],[],[[]]]
 => [[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 1 = 0 + 1
[[],[],[[]],[]]
 => [[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 2 = 1 + 1
[[],[],[[],[]]]
 => [[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => 3 = 2 + 1
[[],[],[[[]]]]
 => [[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1 = 0 + 1
[[],[[]],[],[]]
 => [[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 2 = 1 + 1
[[],[[]],[[]]]
 => [[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 1 = 0 + 1
[[],[[],[]],[]]
 => [[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => 2 = 1 + 1
[[],[[[]]],[]]
 => [[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 2 = 1 + 1
[[],[[],[],[]]]
 => [[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => 3 = 2 + 1
[[],[[],[[]]]]
 => [[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => 1 = 0 + 1
[[],[[[]],[]]]
 => [[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => 3 = 2 + 1
[[],[[[],[]]]]
 => [[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => 4 = 3 + 1
[[],[[[[]]]]]
 => [[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1 = 0 + 1
[[[]],[],[],[]]
 => [[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 2 = 1 + 1
[[[]],[],[[]]]
 => [[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 1 = 0 + 1
[[[]],[[]],[]]
 => [[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 2 = 1 + 1
[[[]],[[],[]]]
 => [[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => 3 = 2 + 1
[[[]],[[[]]]]
 => [[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1 = 0 + 1
[[[],[]],[],[]]
 => [[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 2 = 1 + 1
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 2 = 1 + 1
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => 1 = 0 + 1
[[[[]]],[[]]]
 => [[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1 = 0 + 1
[[[],[],[]],[]]
 => [[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => 2 = 1 + 1
[[[],[[]]],[]]
 => [[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 2 = 1 + 1
[[[[]],[]],[]]
 => [[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => 2 = 1 + 1
[[[[],[]]],[]]
 => [[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 2 = 1 + 1
[[[[[]]]],[]]
 => [[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 2 = 1 + 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Matching statistic: St001184
(load all 2 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => [1,0]
 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[[]]]
 => [1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St000678
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => [1,0]
 => ? = 0 + 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000260
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 33%
Values
[[]]
 => [1,0]
 => [1] => ([],1)
 => 0
[[],[]]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1
[[[]]]
 => [1,1,0,0]
 => [2] => ([],2)
 => ? = 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ? = 0
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => ? = 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => ? = 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => ? = 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => ? = 0
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => ? = 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => ? = 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ? = 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 0
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? = 2
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5] => ([],5)
 => ? = 0
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [5] => ([],5)
 => ? = 2
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => ([],5)
 => ? = 3
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => ([],5)
 => ? = 0
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? = 2
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => ([],5)
 => ? = 0
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => ([],5)
 => ? = 2
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => ? = 2
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => ([],5)
 => ? = 3
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => ([],5)
 => ? = 0
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => ([],5)
 => ? = 3
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => ([],5)
 => ? = 4
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => ? = 0
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 2
[[],[],[[],[[]]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 0
[[],[],[[[]],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 2
[[],[],[[[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 3
[[],[],[[[[]]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 0
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[]],[[],[]],[]]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[],[[]]],[],[]]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[[]],[]],[],[]]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[[],[]]],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.