- FindStat

Your data matches 27 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001913

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of preimages of an integer partition in Bulgarian solitaire. A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row. Partitions without preimages are called garden of eden partitions [1].

Matching statistic: St000260
(load all 19 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => [1,2] => ([],2)
 => ? = 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ? = 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ? = 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ? = 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ? = 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? = 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? = 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ([],6)
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => ([(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [6,2,5,4,3,1] => ([(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)
 => 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [6,4,3,2,5,1] => ([(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)
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [6,5,3,4,2,1] => ([(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)
 => 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,7,5,6,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,6,4,5,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [2,3,7,4,6,5,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,7,5,4,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [2,4,3,7,5,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,5,3,4,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [2,5,3,4,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [2,7,3,4,5,6,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [2,6,3,5,4,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [2,7,3,5,4,6,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [2,7,3,5,6,4,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [2,7,3,6,5,4,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,6,4,3,5,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [2,7,4,3,6,5,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [2,7,4,5,3,6,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,7,4,5,6,3,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [2,7,5,4,3,6,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,7,6,4,5,3,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [3,2,4,7,5,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [3,2,6,4,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [3,2,7,4,6,5,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [3,2,7,5,4,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [4,2,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [4,2,3,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [4,2,3,6,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [4,2,3,7,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [4,2,3,7,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St000772
(load all 3 compositions to match this statistic)

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ? = 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ? = 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ? = 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 1
[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]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ? = 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)
 => ? = 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
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 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)
 => ? = 2
[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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? = 1
[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)
 => ? = 1
[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
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ? = 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)
 => ? = 2
[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)
 => ? = 2
[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
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1
[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
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? = 1
[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)
 => ? = 1
[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)
 => ? = 1
[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
[1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[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)
 => 1
[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)
 => ? = 1
[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)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,5),(1,2),(1,3),(1,4),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(2,5),(3,5),(4,5)],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)
 => 1
[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)
 => 1
[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)
 => 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[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)
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(4,5)],6)
 => ([(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)
 => 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(5,4)],7)
 => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,1),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,5),(1,6),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(6,4)],7)
 => ([(0,5),(1,3),(1,4),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,6),(4,6),(5,6)],7)
 => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6)],7)
 => ([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6)],7)
 => ([(0,4),(1,3),(1,6),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,6),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(1,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6)],7)
 => ([(0,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,2),(1,4),(1,5),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(5,6)],7)
 => ([(0,6),(1,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
 => ([(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6)],7)
 => ([(0,5),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,4),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,6),(1,6),(2,3),(2,4),(2,5),(4,6),(5,6)],7)
 => ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6)],7)
 => ([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(4,5),(4,6)],7)
 => ([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(6,3)],7)
 => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
 => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => 1
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,5),(1,3),(1,4),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 1

Description

The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$0$ , which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$1$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic

$1$ . The graphs with statistic

$n-1$ ,

$n-2$ and

$n-3$ have been characterised, see [1].

Matching statistic: St000259
(load all 13 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => [1,2] => ([],2)
 => ? = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ? = 1 + 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ([],6)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => ([(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [6,2,5,4,3,1] => ([(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
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [6,4,3,2,5,1] => ([(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
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [6,5,3,4,2,1] => ([(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)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,7,5,6,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,6,4,5,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [2,3,7,4,6,5,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,7,5,4,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [2,4,3,7,5,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,5,3,4,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [2,5,3,4,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [2,7,3,4,5,6,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [2,6,3,5,4,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [2,7,3,5,4,6,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [2,7,3,5,6,4,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [2,7,3,6,5,4,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,6,4,3,5,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [2,7,4,3,6,5,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [2,7,4,5,3,6,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,7,4,5,6,3,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [2,7,5,4,3,6,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,7,6,4,5,3,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [3,2,4,7,5,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [3,2,6,4,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [3,2,7,4,6,5,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [3,2,7,5,4,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [4,2,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [4,2,3,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [4,2,3,6,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [4,2,3,7,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [4,2,3,7,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St000456

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => [2,1] => ([],2)
 => ([],2)
 => ? = 1
[1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => ([],3)
 => ? = 1
[1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([],4)
 => ([],4)
 => ? = 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([],5)
 => ([],5)
 => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ([],6)
 => ([],6)
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [4,3,5,1,2,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,5,2,1,3,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,3,1,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,3,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,4,1,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,1,2,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,2,1,3,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,2,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,2,1,7] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,2,1,7] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [5,4,6,2,3,1,7] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [6,5,3,2,4,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [5,6,3,2,4,1,7] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,6,1,7] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [6,3,4,2,5,1,7] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [5,3,4,2,6,1,7] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,1,7] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,1,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [6,4,2,3,5,1,7] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,6,1,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [5,3,2,4,6,1,7] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(6,4)],7)
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,1,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,1,7] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,1,2,7] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,1,2,7] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,1,2,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,1,2,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [6,5,4,2,1,3,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [5,6,4,2,1,3,7] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [6,4,5,2,1,3,7] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [5,4,6,2,1,3,7] => ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [4,5,6,2,1,3,7] => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 1

Description

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

Matching statistic: St001330
(load all 2 compositions to match this statistic)

Mapping

Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => [1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => ([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => ([(0,5),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 2 = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,8,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,7,1,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,6,1,3,4,8,5,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,6,1,3,4,7,8,5] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,5,1,3,8,4,6,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,7,4,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,5,1,3,6,8,4,7] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,5,1,3,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,4,1,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,4,1,7,3,5,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,6,3,7,8,5] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,4,1,5,8,3,6,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,4,1,5,7,3,8,6] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
 => 2 = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,4,1,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,8,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [2,3,7,1,4,5,8,6] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [2,3,6,1,4,8,5,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
 => 2 = 1 + 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: St001208

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => [1]
 => [1,0]
 => [2,1] => 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 1
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ? = 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => [2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [8,1,4,5,2,7,3,6] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [7,3,4,1,6,2,5] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,8,7,3,6] => ? = 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => ? = 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,1,2,8,7,3,5,6] => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => ? = 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,1,2,3,7,4,5,6] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => ? = 2
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 2
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => [10,1,4,6,2,7,9,3,5,8] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => [9,3,5,1,6,8,2,4,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [4,1,2,10,6,7,9,3,5,8] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [3,1,9,5,6,8,2,4,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [2,8,4,5,7,1,3,6] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => [7,1,10,6,2,3,9,4,5,8] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => [6,9,5,1,2,8,3,4,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [10,1,2,7,6,3,9,4,5,8] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [9,1,6,5,2,8,3,4,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [8,5,4,1,7,2,3,6] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [10,1,2,3,6,7,9,4,5,8] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [9,1,2,5,6,8,3,4,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [8,1,4,5,7,2,3,6] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,1,4,5,2,10,9,3,7,8] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => [5,3,4,1,9,8,2,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [4,1,2,5,6,10,9,3,7,8] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [3,1,4,5,9,8,2,6,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [2,3,4,8,7,1,5,6] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => [10,1,6,5,2,3,9,4,7,8] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => [9,5,4,1,2,8,3,6,7] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [10,1,2,5,6,3,9,4,7,8] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [9,1,4,5,2,8,3,6,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [8,3,4,1,7,2,5,6] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [1,0]
 => [2,1] => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => [1,0]
 => [2,1] => 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 1

Description

The number of connected components of the quiver of

$A/T$ when

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

$A$ of

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

Matching statistic: St001236

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => [1]
 => 10 => [1,2] => 1
[1,0,1,0,1,0]
 => [2,1]
 => 1010 => [1,2,2] => 1
[1,0,1,1,0,0]
 => [1,1]
 => 110 => [1,1,2] => 1
[1,1,0,0,1,0]
 => [2]
 => 100 => [1,3] => 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 101010 => [1,2,2,2] => ? = 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 11010 => [1,1,2,2] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => [1,3,1,2] => ? = 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 10110 => [1,2,1,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => [1,1,1,2] => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => [1,2,3] => 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 1100 => [1,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => [1,3,2] => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => [1,4] => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => 10 => [1,2] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 10101010 => [1,2,2,2,2] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 1101010 => [1,1,2,2,2] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 10011010 => [1,3,1,2,2] => ? = 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 1011010 => [1,2,1,2,2] => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 111010 => [1,1,1,2,2] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 10100110 => [1,2,3,1,2] => ? = 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 1100110 => [1,1,3,1,2] => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 10010110 => [1,3,2,1,2] => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 1010110 => [1,2,2,1,2] => ? = 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 110110 => [1,1,2,1,2] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 10001110 => [1,4,1,1,2] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1001110 => [1,3,1,1,2] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 101110 => [1,2,1,1,2] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => [1,1,1,1,2] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1010100 => [1,2,2,3] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 110100 => [1,1,2,3] => ? = 2
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1001100 => [1,3,1,3] => ? = 2
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 101100 => [1,2,1,3] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 11100 => [1,1,1,3] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 1010010 => [1,2,3,2] => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 110010 => [1,1,3,2] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 1001010 => [1,3,2,2] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1000110 => [1,4,1,2] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 10110 => [1,2,1,2] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 101000 => [1,2,4] => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 11000 => [1,1,4] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 100100 => [1,3,3] => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 100010 => [1,4,2] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 10010 => [1,3,2] => 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 110 => [1,1,2] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 10000 => [1,5] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 100 => [1,3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => 1010101010 => [1,2,2,2,2,2] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => 110101010 => [1,1,2,2,2,2] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => 1001101010 => [1,3,1,2,2,2] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => 101101010 => [1,2,1,2,2,2] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => 11101010 => [1,1,1,2,2,2] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => 1010011010 => [1,2,3,1,2,2] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => 110011010 => [1,1,3,1,2,2] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => 1001011010 => [1,3,2,1,2,2] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => 101011010 => [1,2,2,1,2,2] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => 11011010 => [1,1,2,1,2,2] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => 1000111010 => [1,4,1,1,2,2] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => 100111010 => [1,3,1,1,2,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => 10111010 => [1,2,1,1,2,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => 1111010 => [1,1,1,1,2,2] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => 1010100110 => [1,2,2,3,1,2] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => 110100110 => [1,1,2,3,1,2] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => 1001100110 => [1,3,1,3,1,2] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => 101100110 => [1,2,1,3,1,2] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => 11100110 => [1,1,1,3,1,2] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => 1010010110 => [1,2,3,2,1,2] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => 110010110 => [1,1,3,2,1,2] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => 1001010110 => [1,3,2,2,1,2] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => 101010110 => [1,2,2,2,1,2] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => 11010110 => [1,1,2,2,1,2] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => 10110 => [1,2,1,2] => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 1110 => [1,1,1,2] => 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => 1100 => [1,1,3] => 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => 10010 => [1,3,2] => 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => 1010 => [1,2,2] => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => 1000 => [1,4] => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => 10 => [1,2] => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => 11110 => [1,1,1,1,2] => 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => 11100 => [1,1,1,3] => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => 11010 => [1,1,2,2] => 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => 10110 => [1,2,1,2] => 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,3]
 => 11000 => [1,1,4] => 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => 10100 => [1,2,3] => 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [3,1]
 => 10010 => [1,3,2] => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1]
 => 110 => [1,1,2] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => 10000 => [1,5] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => 100 => [1,3] => 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => 10110 => [1,2,1,2] => 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => 1110 => [1,1,1,2] => 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [2,2]
 => 1100 => [1,1,3] => 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [3,1]
 => 10010 => [1,3,2] => 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [3]
 => 1000 => [1,4] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => 10 => [1,2] => 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => 11110 => [1,1,1,1,2] => 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => 10110 => [1,2,1,2] => 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
 => [3,1]
 => 10010 => [1,3,2] => 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [2,1]
 => 1010 => [1,2,2] => 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1]
 => 110 => [1,1,2] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [4]
 => 10000 => [1,5] => 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => 100 => [1,3] => 1

Description

The dominant dimension of the corresponding Comp-Nakayama algebra.

Matching statistic: St001371

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => [1]
 => [1,0]
 => 10 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 1 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => 10 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 10111011000100 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 111011000100 => ? = 0 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => 10101111000100 => ? = 0 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => 10111010010100 => ? = 0 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 111010010100 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => 10101110010100 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 101110010100 => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 10101011010100 => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 101110110000 => ? = 0 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 101110100100 => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => ? = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => 101110111001000100 => ? = 0 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => 1110111001000100 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => 101011111001000100 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => 1011111001000100 => ? = 0 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => 11111001000100 => ? = 0 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => 101110101101000100 => ? = 0 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => 1110101101000100 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => 101011101101000100 => ? = 0 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => 1011101101000100 => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => 11101101000100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => 101010111101000100 => ? = 0 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => 1010111101000100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => 10111101000100 => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 111101000100 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => 101110111000010100 => ? = 0 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => 1110111000010100 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 101011111000010100 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 1011111000010100 => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 11111000010100 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => 101110101100010100 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => 1110101100010100 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => 101011101100010100 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => 1011101100010100 => ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => 11101100010100 => ? = 1 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0 = 1 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0 = 1 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [1,0]
 => 10 => 0 = 1 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0 = 1 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 0 = 1 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0 = 1 - 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => [1,0]
 => 10 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0 = 1 - 1

Description

The length of the longest Yamanouchi prefix of a binary word. This is the largest index

$i$ such that in each of the prefixes

$w_1$ ,

$w_1w_2$ ,

$w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

Matching statistic: St001730

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => [1]
 => [1,0]
 => 10 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 1 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => 10 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 10111011000100 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 111011000100 => ? = 0 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => 10101111000100 => ? = 0 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => 10111010010100 => ? = 0 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 111010010100 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => 10101110010100 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 101110010100 => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 10101011010100 => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 101110110000 => ? = 0 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 101110100100 => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => ? = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => 101110111001000100 => ? = 0 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => 1110111001000100 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => 101011111001000100 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => 1011111001000100 => ? = 0 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => 11111001000100 => ? = 0 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => 101110101101000100 => ? = 0 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => 1110101101000100 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => 101011101101000100 => ? = 0 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => 1011101101000100 => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => 11101101000100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => 101010111101000100 => ? = 0 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => 1010111101000100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => 10111101000100 => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 111101000100 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => 101110111000010100 => ? = 0 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => 1110111000010100 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 101011111000010100 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 1011111000010100 => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 11111000010100 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => 101110101100010100 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => 1110101100010100 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => 101011101100010100 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => 1011101100010100 => ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => 11101100010100 => ? = 1 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0 = 1 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0 = 1 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [1,0]
 => 10 => 0 = 1 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0 = 1 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 0 = 1 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0 = 1 - 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => [1,0]
 => 10 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0 = 1 - 1

Description

The number of times the path corresponding to a binary word crosses the base line. Interpret each

$0$ as a step

$(1,-1)$ and

$1$ as a step

$(1,1)$ . Then this statistic counts the number of times the path crosses the

$x$ -axis.

The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001545The second Elser number of a connected graph. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001625The Möbius invariant of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition.