- FindStat

Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001294
(load all 6 compositions to match this statistic)

Mapping

St001294: 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]
 => 2
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,0]
 => 2
[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]
 => 1
[1,1,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => 3
[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]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => 1
[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]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => 2
[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]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => 1

Description

The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]]. The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers https://oeis.org/A001006.

Matching statistic: St001330
(load all 12 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => ([],2)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,1] => [6,5,4,3,2,1] => ([(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)
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,1,6,5] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,6,1,5] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,5,1,6,4] => [6,5,3,2,1,4] => ([(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)
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,5,6,1,4] => [6,3,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4,6] => [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,6,1,4,5] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,6,3,5] => [2,1,6,4,3,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [2,4,1,5,6,3] => [6,5,4,2,1,3] => ([(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)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [2,4,1,5,3,6] => [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [2,4,5,1,6,3] => [6,5,2,1,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,1,3] => [6,2,1,5,4,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,4,5,1,3,6] => [5,2,1,4,3,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,6,5] => [4,2,1,3,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,5] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,4,6,1,3,5] => [6,2,1,4,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,4,1,3,5,6] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,1,5,3,6,4] => [2,1,6,5,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,1,5,6,3,4] => [2,1,6,3,5,4] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,5,1,3,6,4] => [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,5,1,6,3,4] => [6,2,1,3,5,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,5,6,1,3,4] => [6,2,1,5,3,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => [5,2,1,3,4,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,4,6,2,5] => [1,6,4,3,2,5] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,5,2,6,4] => [1,6,5,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,3,5,6,2,4] => [1,6,3,2,5,4] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4,6] => [1,5,3,2,4,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,3,6,2,4,5] => [1,6,3,2,4,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,4,5,6,2] => [6,5,4,3,1,2] => ([(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)
 => ? = 3 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [3,1,4,5,2,6] => [5,4,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St000454

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => ([],1)
 => ([],1)
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => ([],2)
 => ([(0,1)],2)
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? = 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ([],6)
 => ([(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)
 => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(2,3),(2,4),(2,5)],6)
 => ([(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)
 => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => ([(2,5),(3,4)],6)
 => 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(3,4),(3,5)],6)
 => ([(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)
 => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(5,2),(5,3)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => ([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(2,5),(3,4)],6)
 => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ([(4,5)],6)
 => 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(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)
 => 4
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3

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: St001232
(load all 3 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [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]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001296

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001296: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 3

Description

The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]].

Matching statistic: St000381

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1,0]
 => 10 => [1,2] => 2 = 0 + 2
[1,0,1,0]
 => [1,1,0,0]
 => 1100 => [1,1,3] => 3 = 1 + 2
[1,1,0,0]
 => [1,0,1,0]
 => 1010 => [1,2,2] => 2 = 0 + 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 111000 => [1,1,1,4] => 4 = 2 + 2
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 110100 => [1,1,2,3] => 3 = 1 + 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 110010 => [1,1,3,2] => 3 = 1 + 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 101100 => [1,2,1,3] => 3 = 1 + 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 101010 => [1,2,2,2] => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => [1,1,1,1,5] => 5 = 3 + 2
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => [1,1,1,2,4] => 4 = 2 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => [1,1,1,3,3] => 3 = 1 + 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => [1,1,2,1,4] => 4 = 2 + 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => [1,1,2,2,3] => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => [1,1,1,4,2] => 4 = 2 + 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => [1,1,2,3,2] => 3 = 1 + 2
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => [1,1,3,1,3] => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => [1,2,1,1,4] => 4 = 2 + 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => [1,2,1,2,3] => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => [1,1,3,2,2] => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => [1,2,1,3,2] => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => [1,2,2,1,3] => 3 = 1 + 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => [1,2,2,2,2] => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 4 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => [1,1,1,1,2,5] => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => [1,1,1,1,3,4] => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => [1,1,1,2,1,5] => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => [1,1,1,2,2,4] => ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => [1,1,1,1,4,3] => ? = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => [1,1,1,2,3,3] => ? = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => [1,1,1,3,1,4] => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => [1,1,2,1,1,5] => ? = 3 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => [1,1,2,1,2,4] => ? = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => [1,1,1,3,2,3] => ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => [1,1,2,1,3,3] => ? = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => [1,1,2,2,1,4] => ? = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => [1,1,2,2,2,3] => ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => [1,1,1,1,5,2] => ? = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => [1,1,1,2,4,2] => ? = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => [1,1,1,3,3,2] => ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => [1,1,2,1,4,2] => ? = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => [1,1,2,2,3,2] => ? = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => [1,1,1,4,1,3] => ? = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => [1,1,2,3,1,3] => ? = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => [1,1,3,1,1,4] => ? = 3 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => [1,2,1,1,1,5] => ? = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => [1,2,1,1,2,4] => ? = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => [1,1,3,1,2,3] => ? = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => [1,2,1,1,3,3] => ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => [1,2,1,2,1,4] => ? = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => [1,2,1,2,2,3] => ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => [1,1,1,4,2,2] => ? = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => [1,1,2,3,2,2] => ? = 1 + 2
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => [1,1,3,1,3,2] => ? = 1 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => [1,2,1,1,4,2] => ? = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => [1,2,1,2,3,2] => ? = 1 + 2
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => [1,1,3,2,1,3] => ? = 1 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => [1,2,1,3,1,3] => ? = 1 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => [1,2,2,1,1,4] => ? = 2 + 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => [1,2,2,1,2,3] => ? = 1 + 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => [1,1,3,2,2,2] => ? = 1 + 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => [1,2,1,3,2,2] => ? = 1 + 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => [1,2,2,1,3,2] => ? = 1 + 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => [1,2,2,2,1,3] => ? = 1 + 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => [1,2,2,2,2,2] => ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 111111000000 => [1,1,1,1,1,1,7] => ? = 5 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 111110100000 => [1,1,1,1,1,2,6] => ? = 4 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 111110010000 => [1,1,1,1,1,3,5] => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 111101100000 => [1,1,1,1,2,1,6] => ? = 4 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 111101010000 => [1,1,1,1,2,2,5] => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 111110001000 => [1,1,1,1,1,4,4] => ? = 2 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 111101001000 => [1,1,1,1,2,3,4] => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 111100110000 => [1,1,1,1,3,1,5] => ? = 3 + 2

Description

The largest part of an integer composition.

Matching statistic: St000983

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1,0]
 => 10 => 01 => 2 = 0 + 2
[1,0,1,0]
 => [1,1,0,0]
 => 1100 => 1011 => 3 = 1 + 2
[1,1,0,0]
 => [1,0,1,0]
 => 1010 => 1001 => 2 = 0 + 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 111000 => 010111 => 4 = 2 + 2
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 110100 => 010011 => 3 = 1 + 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 110010 => 011011 => 3 = 1 + 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 101100 => 010001 => 3 = 1 + 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 101010 => 011001 => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 10101111 => 5 = 3 + 2
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 10100111 => 4 = 2 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 10110111 => 3 = 1 + 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 10100011 => 4 = 2 + 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 10110011 => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 10010111 => 4 = 2 + 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 10010011 => 3 = 1 + 2
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 10111011 => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 10100001 => 4 = 2 + 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 10110001 => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 10011011 => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 10010001 => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 10111001 => 3 = 1 + 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 10011001 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 0101011111 => ? = 4 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 0101001111 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0101101111 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0101000111 => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 0101100111 => ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0100101111 => ? = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 0100100111 => ? = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0101110111 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0101000011 => ? = 3 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 0101100011 => ? = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 0100110111 => ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 0100100011 => ? = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 0101110011 => ? = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 0100110011 => ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0110101111 => ? = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 0110100111 => ? = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0110110111 => ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 0110100011 => ? = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 0110110011 => ? = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0100010111 => ? = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 0100010011 => ? = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0101111011 => ? = 3 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0101000001 => ? = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0101100001 => ? = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 0100111011 => ? = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0100100001 => ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0101110001 => ? = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0100110001 => ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0110010111 => ? = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 0110010011 => ? = 1 + 2
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0110111011 => ? = 1 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0110100001 => ? = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0110110001 => ? = 1 + 2
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0100011011 => ? = 1 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0100010001 => ? = 1 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101111001 => ? = 2 + 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0100111001 => ? = 1 + 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 0110011011 => ? = 1 + 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0110010001 => ? = 1 + 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0110111001 => ? = 1 + 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0100011001 => ? = 1 + 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0110011001 => ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 111111000000 => 101010111111 => ? = 5 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 111110100000 => 101010011111 => ? = 4 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 111110010000 => 101011011111 => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 111101100000 => 101010001111 => ? = 4 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 111101010000 => 101011001111 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 111110001000 => 101001011111 => ? = 2 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 111101001000 => 101001001111 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 111100110000 => 101011101111 => ? = 3 + 2

Description

The length of the longest alternating subword. This is the length of the longest consecutive subword of the form

$010...$ or of the form

$101...$ .

Matching statistic: St001183

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001183: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2 = 0 + 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 2 + 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 1 + 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 1 + 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 1 + 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5 = 3 + 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 3 = 1 + 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4 = 2 + 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3 = 1 + 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 3 = 1 + 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 3 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 1 + 2
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 1 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 2
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 1 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 1 + 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 4 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 4 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 3 + 2

Description

The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001207
(load all 5 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1,0]
 => [[1]]
 => [1] => ? = 0
[1,0,1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 1
[1,1,0,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => [3,2,1] => 2
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [1,4,3,2] => 2
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [3,2,1,4] => 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [5,4,3,2,1] => ? = 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,5,4,3,2] => ? = 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
 => [2,1,5,4,3] => ? = 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,5,4,3,2] => ? = 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [3,2,1,5,4] => ? = 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [2,1,5,4,3] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,5,4,3,2] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [4,3,2,1,5] => ? = 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => ? = 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => ? = 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => ? = 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [3,2,1,5,4] => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [2,1,5,4,3] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,5,4,3,2] => ? = 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,2,1,4,5] => ? = 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => ? = 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => ? = 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => [6,5,4,3,2,1] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => [1,6,5,4,3,2] => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0]]
 => [2,1,6,5,4,3] => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,6,5,4,3,2] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,-1,1],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
 => [1,2,6,5,4,3] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[1,0,0,-1,1,0],[0,0,0,1,0,0]]
 => [3,2,1,6,5,4] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
 => [1,3,2,6,5,4] => ? = 2

Description

The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .

Matching statistic: St001624
(load all 2 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%

Values

[1,0]
 => [1,0]
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => ?
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
 => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
 => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => ?
 => ? = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
 => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
 => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
 => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
 => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ([],6)
 => ?
 => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,1),(0,2),(1,7),(2,7),(3,11),(3,12),(3,13),(4,9),(4,10),(4,13),(5,8),(5,10),(5,12),(6,8),(6,9),(6,11),(7,3),(7,4),(7,5),(7,6),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18)],19)
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4),(1,5)],6)
 => ?
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => ([(0,5),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,6),(6,1),(6,2),(6,3),(6,4),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17)],18)
 => ? = 3 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1

Description

The breadth of a lattice. The '''breadth''' of a lattice is the least integer

$b$ such that any join

$x_1\vee x_2\vee\cdots\vee x_n$ , with

$n > b$ , can be expressed as a join over a proper subset of

$\{x_1,x_2,\ldots,x_n\}$ .