- FindStat

Your data matches 22 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001879
(load all 34 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

Matching statistic: St001232
(load all 9 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 57%●distinct values known / distinct values provided: 56%

Values

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

Description

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

Matching statistic: St001880
(load all 33 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 56% ●values known / values provided: 57%●distinct values known / distinct values provided: 56%

Values

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

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Matching statistic: St000454

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 47%●distinct values known / distinct values provided: 11%

Values

[3,1,2] => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2 - 4
[3,2,1] => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2 - 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4 - 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 4
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 6 - 4
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 5 - 4
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 5 - 4
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 5 - 4
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 6 - 4
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 6 - 4
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 5 - 4
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 6 - 4
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 6 - 4
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 - 4
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 8 - 4
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[.,[.,[[.,.],[.,.]]]],.]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 7 - 4
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[.,[.,[[.,.],[.,.]]]],.]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 7 - 4
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[.,[[[.,.],[.,.]],.]],.]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 7 - 4
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[.,[[[.,.],[.,.]],.]],.]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 7 - 4
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],[.,.]]],.]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 6 - 4
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],[.,.]]],.]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 6 - 4
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],[.,.]]],.]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 6 - 4
[7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,3,4,6,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,3,5,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,3,5,6,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,3,6,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,3,6,5,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,4,3,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,4,3,6,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,4,5,3,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,4,5,6,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,4,6,3,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,4,6,5,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,5,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,5,3,6,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,5,4,3,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,5,4,6,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,5,6,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,5,6,4,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,6,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,6,3,5,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,6,4,3,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,6,4,5,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,6,5,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,2,6,5,4,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,2,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,2,4,6,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,2,5,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,2,5,6,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,2,6,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,2,6,5,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,4,2,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,4,2,6,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,4,5,2,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,4,5,6,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,4,6,2,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,4,6,5,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,5,2,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,5,2,6,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,5,4,2,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,5,4,6,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,5,6,2,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,5,6,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,6,2,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,6,2,5,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,6,4,2,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,6,4,5,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,6,5,2,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,3,6,5,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,4,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4
[7,1,4,2,3,6,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 6 - 4

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

Matching statistic: St000718

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 44%

Values

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

Description

The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]

Matching statistic: St000422
(load all 2 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 11%

Values

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

Description

The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph

$K_n$ equals

$2n-2$ . For this reason, we do not define the energy of the empty graph.

Matching statistic: St000912
(load all 2 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 56%

Values

[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 5 = 4 + 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4 = 3 + 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4 = 3 + 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 6 + 1
[2,1,5,3,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 5 + 1
[2,1,5,4,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 5 + 1
[3,1,2,5,4] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 5 + 1
[3,1,5,2,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 6 + 1
[3,1,5,4,2] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 6 + 1
[3,2,1,5,4] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 5 + 1
[3,2,5,1,4] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 6 + 1
[3,2,5,4,1] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 6 + 1
[5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,2,4,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,3,2,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,3,4,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,2,1,3,4] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,2,1,4,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,2,3,1,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 4 + 1
[5,2,3,4,1] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 4 + 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 4 + 1
[5,2,4,3,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 4 + 1
[5,3,1,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5 = 4 + 1
[5,3,1,4,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5 = 4 + 1
[5,3,2,1,4] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,2,4,1] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,4,1,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,4,2,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,1,2,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5 = 4 + 1
[5,4,1,3,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5 = 4 + 1
[5,4,2,1,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,2,3,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,3,1,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[2,1,3,4,6,5] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 8 + 1
[2,1,3,6,4,5] => [5,6,4,1,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 7 + 1
[2,1,3,6,5,4] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 7 + 1
[2,1,4,6,3,5] => [5,6,3,1,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 7 + 1
[2,1,4,6,5,3] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 7 + 1
[2,1,6,3,4,5] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,3,5,4] => [5,6,1,4,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,4,3,5] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,4,5,3] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,5,3,4] => [5,6,1,2,4,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,5,4,3] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[3,1,2,4,6,5] => [4,6,5,3,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,2,6,4,5] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 + 1
[3,1,2,6,5,4] => [4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 + 1
[3,1,4,2,6,5] => [4,6,3,5,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,4,6,2,5] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 8 + 1
[3,1,4,6,5,2] => [4,6,3,1,2,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 8 + 1
[3,1,6,2,4,5] => [4,6,1,5,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,2,5,4] => [4,6,1,5,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,4,2,5] => [4,6,1,3,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,4,5,2] => [4,6,1,3,2,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,5,2,4] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,5,4,2] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,2,1,4,6,5] => [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 7 + 1
[3,2,1,6,4,5] => [4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 6 + 1
[3,2,1,6,5,4] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 6 + 1
[3,2,4,1,6,5] => [4,5,3,6,1,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 7 + 1
[3,2,4,6,1,5] => [4,5,3,1,6,2] => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 8 + 1
[3,2,4,6,5,1] => [4,5,3,1,2,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 8 + 1
[3,2,6,1,4,5] => [4,5,1,6,3,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 7 + 1
[3,2,6,1,5,4] => [4,5,1,6,2,3] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 7 + 1
[3,2,6,4,1,5] => [4,5,1,3,6,2] => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 7 + 1
[3,2,6,4,5,1] => [4,5,1,3,2,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 7 + 1
[6,4,2,3,1,5] => [1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,4,2,3,5,1] => [1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,4,2,5,1,3] => [1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,4,2,5,3,1] => [1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,4,3,2,1,5] => [1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,3,2,5,1] => [1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,3,5,1,2] => [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,3,5,2,1] => [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,5,2,1,3] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,5,3,1,2] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,5,3,2,1] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,2,3,1,4] => [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,5,2,3,4,1] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,5,2,4,1,3] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,5,2,4,3,1] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,5,3,2,1,4] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,3,2,4,1] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,3,4,1,2] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,3,4,2,1] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,4,2,1,3] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,4,2,3,1] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,4,3,1,2] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[7,5,4,3,2,1,6] => [1,3,4,5,6,7,2] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 6 + 1

Description

The number of maximal antichains in a poset.

Matching statistic: St001631
(load all 2 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001631: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 56%

Values

[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 4
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 3
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 3
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,1,3,5,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 6
[2,1,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 5
[2,1,5,4,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 5
[3,1,2,5,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 5
[3,1,5,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 6
[3,1,5,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 6
[3,2,1,5,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 5
[3,2,5,1,4] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 6
[3,2,5,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 6
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[5,1,2,4,3] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 4
[5,1,3,2,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 4
[5,1,3,4,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 4
[5,1,4,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 4
[5,1,4,3,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 4
[5,2,1,3,4] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 4
[5,2,1,4,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 4
[5,2,3,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 4
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[5,2,4,1,3] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 4
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 4
[5,3,1,2,4] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 4
[5,3,1,4,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 4
[5,3,2,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 4
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 4
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[5,4,1,3,2] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 4
[5,4,2,1,3] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 4
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[2,1,3,4,6,5] => [1,3,4,6,2,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
 => ? = 8
[2,1,3,6,4,5] => [1,3,6,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
 => ? = 7
[2,1,3,6,5,4] => [1,3,6,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
 => ? = 7
[2,1,4,6,3,5] => [1,4,6,2,3,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
 => ? = 7
[2,1,4,6,5,3] => [1,4,6,2,3,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
 => ? = 7
[2,1,6,3,4,5] => [1,6,2,3,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 6
[2,1,6,3,5,4] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 6
[2,1,6,4,3,5] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
 => ? = 6
[2,1,6,4,5,3] => [1,6,2,3,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 6
[2,1,6,5,3,4] => [1,6,2,3,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 6
[2,1,6,5,4,3] => [1,6,2,3,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 6
[3,1,2,4,6,5] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
 => ? = 7
[3,1,2,6,4,5] => [1,2,6,3,4,5] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 6
[3,1,2,6,5,4] => [1,2,6,3,4,5] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 6
[3,1,4,2,6,5] => [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
 => ? = 7
[3,1,4,6,2,5] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
 => ? = 8
[3,1,4,6,5,2] => [1,4,6,2,3,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
 => ? = 8
[3,1,6,2,4,5] => [1,6,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 7
[3,1,6,2,5,4] => [1,6,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 7
[3,1,6,4,2,5] => [1,6,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 7
[3,1,6,4,5,2] => [1,6,2,3,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 7
[3,1,6,5,2,4] => [1,6,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 7
[6,1,2,3,4,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,2,3,4,5,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,3,4,5,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,3,4,5,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,4,5,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,4,5,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,4,5,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,4,5,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,5,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,5,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,5,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,5,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,5,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,5,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,5,4,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[7,1,2,3,4,5,6] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,2,3,4,5,6,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,3,4,5,6,1,2] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,3,4,5,6,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,4,5,6,1,2,3] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,4,5,6,2,3,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,4,5,6,3,1,2] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,5,6,1,2,3,4] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,5,6,2,3,4,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,5,6,3,4,1,2] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,5,6,4,1,2,3] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,5,6,4,2,3,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,5,6,4,3,1,2] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,6,1,2,3,4,5] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,6,2,3,4,5,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,6,3,4,5,1,2] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[7,6,3,4,5,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6

Description

The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset.

Matching statistic: St001330

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 56%

Values

[3,1,2] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 1
[2,1,5,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 1
[2,1,5,4,3] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 1
[3,1,2,5,4] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 1
[3,1,5,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 1
[3,1,5,4,2] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 1
[3,2,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 1
[3,2,5,1,4] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 1
[3,2,5,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 1
[5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5,1,2,4,3] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,3,2,4] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,3,4,2] => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,4,2,3] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,4,3,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,2,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,2,1,4,3] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,2,3,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5,2,4,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,2,4,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,3,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,3,1,4,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,3,2,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,3,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5,4,1,3,2] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,4,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,1,3,4,6,5] => [1,3,4,6,2,5] => [5,2,6,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8 + 1
[2,1,3,6,4,5] => [1,3,6,2,4,5] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[2,1,3,6,5,4] => [1,3,6,2,4,5] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[2,1,4,6,3,5] => [1,4,6,2,3,5] => [5,3,2,6,4,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[2,1,4,6,5,3] => [1,4,6,2,3,5] => [5,3,2,6,4,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[2,1,6,3,4,5] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 + 1
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 + 1
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 + 1
[2,1,6,4,5,3] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 + 1
[2,1,6,5,3,4] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 + 1
[2,1,6,5,4,3] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 + 1
[3,1,2,4,6,5] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(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)
 => ? = 7 + 1
[3,1,2,6,4,5] => [1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(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)
 => ? = 6 + 1
[3,1,2,6,5,4] => [1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(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)
 => ? = 6 + 1
[3,1,4,2,6,5] => [1,4,2,6,3,5] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[3,1,4,6,2,5] => [1,4,6,2,5,3] => [3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 8 + 1
[3,1,4,6,5,2] => [1,4,6,2,3,5] => [5,3,2,6,4,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8 + 1
[3,1,6,2,4,5] => [1,6,2,4,5,3] => [3,5,4,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[3,1,6,2,5,4] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[3,1,6,4,2,5] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[3,1,6,4,5,2] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[3,1,6,5,2,4] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 + 1
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,3,4,5,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,3,4,5,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,4,5,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,4,5,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,4,5,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,4,5,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,5,1,2,3,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,5,2,3,4,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,5,3,4,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,5,3,4,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,5,4,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,5,4,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,5,4,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6,5,4,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[7,1,2,3,4,5,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,2,3,4,5,6,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,3,4,5,6,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,3,4,5,6,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,4,5,6,1,2,3] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,4,5,6,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,4,5,6,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,5,6,1,2,3,4] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,5,6,2,3,4,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,5,6,3,4,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,5,6,4,1,2,3] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,5,6,4,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,5,6,4,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,6,1,2,3,4,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,6,2,3,4,5,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,6,3,4,5,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 1
[7,6,3,4,5,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7 = 6 + 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: St000680
(load all 2 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000680: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 44%

Values

[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 5 = 4 + 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4 = 3 + 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4 = 3 + 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 6 + 1
[2,1,5,3,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 5 + 1
[2,1,5,4,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 5 + 1
[3,1,2,5,4] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 5 + 1
[3,1,5,2,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 6 + 1
[3,1,5,4,2] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 6 + 1
[3,2,1,5,4] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 5 + 1
[3,2,5,1,4] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 6 + 1
[3,2,5,4,1] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 6 + 1
[5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,2,4,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,3,2,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,3,4,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,2,1,3,4] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,2,1,4,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 4 + 1
[5,2,3,1,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 4 + 1
[5,2,3,4,1] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 4 + 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 4 + 1
[5,2,4,3,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 4 + 1
[5,3,1,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5 = 4 + 1
[5,3,1,4,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5 = 4 + 1
[5,3,2,1,4] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,2,4,1] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,4,1,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,4,2,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,1,2,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5 = 4 + 1
[5,4,1,3,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5 = 4 + 1
[5,4,2,1,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,2,3,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,3,1,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[2,1,3,4,6,5] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 8 + 1
[2,1,3,6,4,5] => [5,6,4,1,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 7 + 1
[2,1,3,6,5,4] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 7 + 1
[2,1,4,6,3,5] => [5,6,3,1,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 7 + 1
[2,1,4,6,5,3] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 7 + 1
[2,1,6,3,4,5] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,3,5,4] => [5,6,1,4,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,4,3,5] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,4,5,3] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,5,3,4] => [5,6,1,2,4,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[2,1,6,5,4,3] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 + 1
[3,1,2,4,6,5] => [4,6,5,3,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,2,6,4,5] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 + 1
[3,1,2,6,5,4] => [4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 + 1
[3,1,4,2,6,5] => [4,6,3,5,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,4,6,2,5] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 8 + 1
[3,1,4,6,5,2] => [4,6,3,1,2,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 8 + 1
[3,1,6,2,4,5] => [4,6,1,5,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,2,5,4] => [4,6,1,5,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,4,2,5] => [4,6,1,3,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,4,5,2] => [4,6,1,3,2,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,5,2,4] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,1,6,5,4,2] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 7 + 1
[3,2,1,4,6,5] => [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 7 + 1
[3,2,1,6,4,5] => [4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 6 + 1
[3,2,1,6,5,4] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 6 + 1
[3,2,4,1,6,5] => [4,5,3,6,1,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 7 + 1
[3,2,4,6,1,5] => [4,5,3,1,6,2] => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 8 + 1
[3,2,4,6,5,1] => [4,5,3,1,2,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 8 + 1
[3,2,6,1,4,5] => [4,5,1,6,3,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 7 + 1
[3,2,6,1,5,4] => [4,5,1,6,2,3] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 7 + 1
[3,2,6,4,1,5] => [4,5,1,3,6,2] => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 7 + 1
[3,2,6,4,5,1] => [4,5,1,3,2,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 7 + 1
[6,4,2,3,1,5] => [1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,4,2,3,5,1] => [1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,4,2,5,1,3] => [1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,4,2,5,3,1] => [1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,4,3,2,1,5] => [1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,3,2,5,1] => [1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,3,5,1,2] => [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,3,5,2,1] => [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,5,2,1,3] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,5,3,1,2] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,4,5,3,2,1] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,2,3,1,4] => [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,5,2,3,4,1] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,5,2,4,1,3] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,5,2,4,3,1] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 5 + 1
[6,5,3,2,1,4] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,3,2,4,1] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,3,4,1,2] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,3,4,2,1] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,4,2,1,3] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,4,2,3,1] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,4,3,1,2] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1

Description

The Grundy value for Hackendot on posets. Two players take turns and remove an order filter. The player who is faced with the one element poset looses. This game is a slight variation of Chomp. This statistic is the Grundy value of the poset, that is, the smallest non-negative integer which does not occur as value of a poset obtained by a single move.

The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000909The number of maximal chains of maximal size in a poset. St000080The rank of the poset. St000189The number of elements in the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001645The pebbling number of a connected graph. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset.