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Mp00051: Ordered trees to Dyck pathDyck paths
St001239: 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]
=> 2
[[[]]]
=> [1,1,0,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> 3
[[[[]]]]
=> [1,1,1,0,0,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 3
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 4
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001192
Mp00048: Ordered trees left-right symmetryOrdered trees
Mp00051: Ordered trees to Dyck pathDyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [1,0]
=> 0 = 1 - 1
[[],[]]
=> [[],[]]
=> [1,0,1,0]
=> 1 = 2 - 1
[[[]]]
=> [[[]]]
=> [1,1,0,0]
=> 0 = 1 - 1
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[[]],[]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[[],[]]]
=> [[[],[]]]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[[[]]]]
=> [[[[]]]]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[]],[]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[[]],[],[]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[[]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[[[]]],[]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[],[],[]]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[[],[[]]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[[[[],[]]]]
=> [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[[[[[]]]]]
=> [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
St000485: Permutations ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => [1] => ? = 1
[[],[]]
=> [1,0,1,0]
=> [2,1] => [2,1] => 2
[[[]]]
=> [1,1,0,0]
=> [1,2] => [1,2] => 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 3
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 3
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 4
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 4
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => 3
Description
The length of the longest cycle of a permutation.