searching the database
Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001192
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001192: 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]
=> 1
[[[]],[]]
=> [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]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [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]
=> 2
[[[[]],[]]]
=> [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]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[],[[],[]],[]]
=> [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]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[[[]],[]],[]]
=> [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]
=> 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001239
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [1,0]
=> 1 = 0 + 1
[[],[]]
=> [[],[]]
=> [1,0,1,0]
=> 2 = 1 + 1
[[[]]]
=> [[[]]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[[]],[]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[[],[]]]
=> [[[],[]]]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[[[[]]]]
=> [[[[]]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[[]],[]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[[]],[],[]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[[]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[[[]]],[]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[],[],[]]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[[],[[]]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[[[],[]]]]
=> [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[[[[]]]]]
=> [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St000444
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
St000444: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => [1,0]
=> ? = 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 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]
=> 2 = 1 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 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]
=> 3 = 2 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 3 = 2 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 3 = 2 + 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]
=> 3 = 2 + 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]
=> 2 = 1 + 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]
=> 3 = 2 + 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]
=> 3 = 2 + 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]
=> 3 = 2 + 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]
=> 4 = 3 + 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
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
Description
The length of the maximal rise of a Dyck path.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!