Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000480
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => []
 => []
 => []
 => 0
[[]]
 => [1,0]
 => []
 => []
 => 0
[[],[]]
 => [1,0,1,0]
 => [1]
 => [1]
 => 0
[[[]]]
 => [1,1,0,0]
 => []
 => []
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1]
 => [3]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => [2]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => [1,1]
 => 0
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1]
 => [1]
 => 0
[[[[]]]]
 => [1,1,1,0,0,0]
 => []
 => []
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [5,1]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,2,1]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [4,1]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [2,2]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [2,1]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [5]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [4]
 => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [2,1,1]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1]
 => 0
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [3]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [2]
 => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [1,1]
 => 0
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [1]
 => 0
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => []
 => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [7,3]
 => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,2,2]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [6,3]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [3,2,2,1]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [4,1,1,1]
 => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [7,2]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [6,2]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [4,4]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [3,2,1,1]
 => 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [5,2]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [4,1,1]
 => 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [3,3]
 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [3,1]
 => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [7,1,1]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [2,2,2,2]
 => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [6,1,1]
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2,2,1]
 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2,2]
 => 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [7,1]
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [7]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [6,1]
 => 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [6]
 => 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [5,1,1]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [3,1,1,1]
 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2,2,1,1]
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [2,1,1,1]
 => 1
Description
The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St001195
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[]
 => []
 => []
 => []
 => ? = 0
[[]]
 => [1,0]
 => []
 => []
 => ? = 0
[[],[]]
 => [1,0,1,0]
 => [1]
 => [1,0]
 => ? = 0
[[[]]]
 => [1,1,0,0]
 => []
 => []
 => ? = 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => ? = 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => ? = 0
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1]
 => [1,0]
 => ? = 0
[[[[]]]]
 => [1,1,1,0,0,0]
 => []
 => []
 => ? = 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => ? = 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => ? = 0
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => ? = 0
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? = 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => ? = 1
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,0,1,0]
 => ? = 0
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0]
 => ? = 0
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? = 0
[[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 2
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 2
[[],[],[[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 2
[[],[],[[],[[]]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 2
[[],[],[[[]],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[[],[],[[[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 2
[[],[],[[[[]]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 2
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 2
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 2
[[],[[],[]],[[]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 2
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 1
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 2
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[[],[[],[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[[],[[],[],[[]]]]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[[],[[[[[]]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[[]],[[[[]]]]]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[[[]]],[[[]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[[[[]]]],[[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[[],[],[[[]]]]]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[[],[[],[[]]]]]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[[[],[[[],[]]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[[],[[[[]]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[[[[]],[],[[]]]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[[[[]],[[],[]]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[[[]],[[[]]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 1
[[[[[]]],[],[]]]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[[[],[]],[[]]]]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[[[[[]]],[[]]]]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1
[[[[[]],[]],[]]]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St000628
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
[]
 => []
 => => ? => ? = 0 + 1
[[]]
 => [1,0]
 => 10 => 10 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => 1010 => 0110 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => 1100 => 1010 => 1 = 0 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 101010 => 100110 => 2 = 1 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 101100 => 110010 => 2 = 1 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 110010 => 010110 => 1 = 0 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 110100 => 011010 => 1 = 0 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 111000 => 101010 => 1 = 0 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 01100110 => 2 = 1 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 01110010 => 2 = 1 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 10100110 => 2 = 1 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 10110010 => 2 = 1 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 11010010 => 2 = 1 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 00110110 => 2 = 1 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 00111010 => 2 = 1 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 10010110 => 2 = 1 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 01010110 => 1 = 0 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 10011010 => 2 = 1 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 11001010 => 2 = 1 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 01011010 => 1 = 0 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 01101010 => 1 = 0 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 10101010 => 1 = 0 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? => ? = 2 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? => ? = 1 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? => ? = 2 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? => ? = 2 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? => ? = 1 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => ? => ? = 2 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? => ? = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? => ? = 1 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => ? => ? = 2 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? => ? = 2 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? => ? = 1 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? => ? = 1 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? => ? = 1 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? => ? = 1 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => ? => ? = 1 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => ? => ? = 1 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => ? => ? = 1 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => ? => ? = 1 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? => ? = 1 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => ? => ? = 1 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => ? => ? = 1 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => ? => ? = 1 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? => ? = 1 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => ? => ? = 1 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? => ? = 1 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? => ? = 1 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? => ? = 1 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? => ? = 0 + 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => ? => ? = 1 + 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => ? => ? = 1 + 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => ? => ? = 1 + 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => ? => ? = 1 + 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => ? => ? = 1 + 1
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? => ? = 1 + 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => ? => ? = 1 + 1
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? => ? = 1 + 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? => ? = 0 + 1
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => ? => ? = 1 + 1
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? => ? = 1 + 1
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => ? => ? = 0 + 1
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => ? => ? = 0 + 1
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => ? => ? = 0 + 1
[[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? => ? = 2 + 1
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 101011100010 => ? => ? = 2 + 1
[[],[],[[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => ? => ? = 2 + 1
[[],[],[[],[[]]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 101011011000 => ? => ? = 2 + 1
[[],[],[[[]],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => ? => ? = 1 + 1
[[],[],[[[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => ? => ? = 2 + 1
[[],[],[[[[]]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => ? => ? = 1 + 1
Description
The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.