Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001928
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => []
 => [1] => [1] => 0
[[]]
 => [1,0]
 => [2,1] => [2,1] => 1
[[],[]]
 => [1,0,1,0]
 => [3,1,2] => [3,1,2] => 1
[[[]]]
 => [1,1,0,0]
 => [2,3,1] => [3,2,1] => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [3,1,4,2] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [4,3,2,1] => 2
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,4,1,2,3] => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [4,1,2,5,3] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,4,3,1,2] => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,5,2,4] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,1,5,2,3] => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [3,1,5,4,2] => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,2,1,5,3] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [4,3,1,5,2] => 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => 2
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,5,1,2,3,4] => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,4,1,2,3,5] => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [5,1,2,3,6,4] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,5,4,1,2,3] => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,3,1,2,4,5] => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,5,3,1,2,4] => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,1,2,6,3,5] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,4,3,1,2,5] => 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [5,1,2,6,3,4] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [4,1,2,6,5,3] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [5,3,1,2,6,4] => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [5,4,1,2,6,3] => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,5,4,3,1,2] => 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6,5,2,1,3,4] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6,4,2,1,3,5] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [5,2,1,3,6,4] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,5,4,2,1,3] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [3,1,6,2,4,5] => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [6,3,2,1,4,5] => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [3,1,6,5,2,4] => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [6,5,3,2,1,4] => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,1,6,2,3,5] => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,1,6,4,2,5] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [4,2,1,6,3,5] => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [4,3,1,6,2,5] => 2
Description
The number of non-overlapping descents in a permutation. In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
Mapping
Mp00246: Ordered trees —rotate⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[]
 => []
 => ([],1)
 => ([],1)
 => ? = 0 - 1
[[]]
 => [[]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 - 1
[[],[]]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 0 = 1 - 1
[[[]]]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 0 = 1 - 1
[[],[],[]]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[],[[]]]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 - 1
[[[]],[]]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[],[]]]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 1
[[[[]]]]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 1
[[],[],[],[]]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[],[[]]]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[],[[]],[]]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[],[[],[]]]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[],[[[]]]]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[[]],[],[]]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[]],[[]]]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[[],[]],[]]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[[[]]],[]]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[[],[],[]]]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[[],[[]]]]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[[[]],[]]]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[[[],[]]]]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[[[[]]]]]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[],[],[],[],[]]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[[],[],[[]],[]]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[[],[],[[],[]]]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[],[[]],[],[]]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[[],[[]],[[]]]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[],[[],[]],[]]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[],[[],[],[]]]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[],[[],[[]]]]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[],[[[]],[]]]
 => [[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[],[[[],[]]]]
 => [[[[],[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[],[[[[]]]]]
 => [[[[[]]]],[]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[]],[],[],[]]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[]],[[]],[]]
 => [[[[]],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[]],[[],[]]]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [[[[[]]],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[],[]],[],[]]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[[]]],[],[]]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[],[]],[[]]]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[[]]],[[]]]
 => [[[[[]],[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[],[],[]],[]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[[]],[]],[]]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[[[]]]],[]]
 => [[[[[],[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[],[],[],[]]]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[],[],[[]]]]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[],[[]],[]]]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[[[],[[],[]]]]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[],[[[]]]]]
 => [[[[]]],[[]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[[[[]],[],[]]]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[[]],[[]]]]
 => [[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[[],[]],[]]]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[[[]]],[]]]
 => [[[[],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[[[[],[],[]]]]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[[[[],[[]]]]]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[[[[[]],[]]]]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
Mapping
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 75%
Values
[]
 => []
 => ([],1)
 => 0
[[]]
 => [[]]
 => ([(0,1)],2)
 => 1
[[],[]]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => ? = 1
[[[]]]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => ? = 1
[[],[],[]]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[[],[[]]]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1
[[[]],[]]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[[[],[]]]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[[[[]]]]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 2
[[],[],[],[]]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1
[[],[],[[]]]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1
[[],[[]],[]]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1
[[],[[],[]]]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2
[[],[[[]]]]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[[]],[],[]]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1
[[[]],[[]]]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2
[[[],[]],[]]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
[[[[]]],[]]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2
[[[],[],[]]]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
[[[],[[]]]]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2
[[[[]],[]]]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2
[[[[],[]]]]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2
[[[[[]]]]]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[],[],[],[],[]]
 => [[[[[[]]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[[],[],[],[[]]]
 => [[[[[],[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 1
[[],[],[[]],[]]
 => [[[[],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 1
[[],[],[[],[]]]
 => [[[[[]],[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2
[[],[],[[[]]]]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[[],[[]],[],[]]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 1
[[],[[]],[[]]]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[[],[[],[]],[]]
 => [[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2
[[],[[[]]],[]]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[[],[[],[],[]]]
 => [[[[[]]],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2
[[],[[],[[]]]]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[[],[[[]],[]]]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[[],[[[],[]]]]
 => [[[[]],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[[],[[[[]]]]]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[[]],[],[],[]]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[[[]],[],[[]]]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2
[[[]],[[]],[]]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2
[[[]],[[],[]]]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2
[[[]],[[[]]]]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[[[],[]],[],[]]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2
[[[[]]],[[]]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[[[],[],[]],[]]
 => [[[[]]],[[]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2
[[[],[[]]],[]]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2
[[[[]],[]],[]]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2
[[[[],[]]],[]]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2
[[[[[]]]],[]]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[[[],[],[],[]]]
 => [[[[[]]]],[]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2
[[[],[],[[]]]]
 => [[[[],[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2
[[[],[[]],[]]]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3
[[[],[[],[]]]]
 => [[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2
[[[],[[[]]]]]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 3
[[[[]],[],[]]]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2
[[[[]],[[]]]]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[[[[],[[]]]]]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000264
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 25%
Values
[]
 => []
 => [] => ([],0)
 => ? = 0 + 2
[[]]
 => [1,0]
 => [1] => ([],1)
 => ? = 1 + 2
[[],[]]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ? = 1 + 2
[[[]]]
 => [1,1,0,0]
 => [1,2] => ([],2)
 => ? = 1 + 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ? = 1 + 2
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 1 + 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ? = 1 + 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ? = 2 + 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ? = 2 + 2
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? = 1 + 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 2 + 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 + 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4 = 2 + 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? = 2 + 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 + 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? = 2 + 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 2 + 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 2 + 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ? = 2 + 2
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 1 + 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1 + 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4 = 2 + 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1 + 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 2 + 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 2 + 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 2 + 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? = 2 + 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 4 = 2 + 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4 = 2 + 2
[[[]],[[[]]]]
 => [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)
 => 4 = 2 + 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4 = 2 + 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4 = 2 + 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4 = 2 + 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 2 + 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? = 2 + 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 2 + 2
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? = 2 + 2
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 3 + 2
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? = 2 + 2
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 + 2
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? = 2 + 2
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 4 = 2 + 2
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? = 2 + 2
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St001292
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Values
[]
 => .
 => ?
 => ?
 => ? = 0 - 1
[[]]
 => [.,.]
 => [[],[]]
 => [1,0,1,0]
 => 0 = 1 - 1
[[],[]]
 => [.,[.,.]]
 => [[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[[[]]]
 => [[.,.],.]
 => [[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 1 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[],[]]]
 => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[[[]]]]
 => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 1 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 1 - 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 1 - 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => ? = 2 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => ? = 2 - 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [[[],[[],[[],[]]]],[]]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => ? = 1 - 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [[[],[[[],[]],[]]],[]]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => ? = 2 - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [[[[],[[],[]]],[]],[]]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 2 - 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 2 - 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [[[[],[]],[[],[]]],[]]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
 => ? = 2 - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [[[[[],[]],[]],[]],[]]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
 => ? = 2 - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [[],[[],[[],[[],[[],[]]]]]]
 => [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 1 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [[],[[],[[],[[[],[]],[]]]]]
 => [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? = 1 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [[],[[],[[[],[[],[]]],[]]]]
 => [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
 => ? = 1 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [[],[[],[[[],[]],[[],[]]]]]
 => [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [[],[[],[[[[],[]],[]],[]]]]
 => [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
 => ? = 2 - 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [[],[[[],[[],[[],[]]]],[]]]
 => [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 1 - 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [[],[[[],[[[],[]],[]]],[]]]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => ? = 2 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [[],[[[],[[],[]]],[[],[]]]]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
 => ? = 2 - 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [[],[[[[],[[],[]]],[]],[]]]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
 => ? = 2 - 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [[],[[[],[]],[[],[[],[]]]]]
 => [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [[],[[[],[]],[[[],[]],[]]]]
 => [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 2 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [[],[[[[],[]],[]],[[],[]]]]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
 => ? = 2 - 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [[],[[[[],[]],[[],[]]],[]]]
 => [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 2 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [[],[[[[[],[]],[]],[]],[]]]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => ? = 2 - 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [[[],[[],[[],[[],[]]]]],[]]
 => [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 1 - 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [[[],[[],[[[],[]],[]]]],[]]
 => [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 - 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [[[],[[[],[[],[]]],[]]],[]]
 => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => ? = 2 - 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [[[],[]],[[[],[]],[[],[]]]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => ? = 2 - 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [[[],[[[[],[]],[]],[]]],[]]
 => [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
 => ? = 2 - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[],[[],[]]],[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 2 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [[[[],[[],[[],[]]]],[]],[]]
 => [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [[[],[[],[]]],[[[],[]],[]]]
 => [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 2 - 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [[[[],[[[],[]],[]]],[]],[]]
 => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[[],[[],[[],[]]]],[[],[]]]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [[[],[[[],[]],[]]],[[],[]]]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [[[[],[[],[]]],[]],[[],[]]]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [[[[],[[],[]]],[[],[]]],[]]
 => [1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0]
 => ? = 2 - 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [[[[[],[[],[]]],[]],[]],[]]
 => [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 2 - 1
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[],[]],[[],[[],[[],[]]]]]
 => [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 1
[[[],[],[[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [[[],[]],[[],[[[],[]],[]]]]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 2 - 1
[[[],[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [[[],[]],[[[],[[],[]]],[]]]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 3 - 1
[[[],[[],[]]]]
 => [[.,[[.,.],[.,.]]],.]
 => [[[],[[[],[]],[[],[]]]],[]]
 => [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
 => ? = 2 - 1
[[[],[[[]]]]]
 => [[.,.],[[[.,.],.],.]]
 => [[[],[]],[[[[],[]],[]],[]]]
 => [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => ? = 3 - 1
[[[[]],[],[]]]
 => [[[.,.],.],[.,[.,.]]]
 => [[[[],[]],[]],[[],[[],[]]]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 2 - 1
[[[[]],[[]]]]
 => [[[.,.],.],[[.,.],.]]
 => [[[[],[]],[]],[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 2 - 1
Description
The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Here $A$ is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].