- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001632

Mapping

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.

Matching statistic: St001624

Mapping

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%

Values

[[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 1 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 1 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,-1,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2

Description

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

$b$ such that any join

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

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

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

Matching statistic: St001630

Mapping

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%

Values

[[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 1 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 1 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,-1,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2

Description

The global dimension of the incidence algebra of the lattice over the rational numbers.

Matching statistic: St001878

Mapping

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%

Values

[[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 1 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 1 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 1 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 0 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,-1,1,0],[0,0,1,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2

Description

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.

Matching statistic: St000664

Mapping

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of right ropes of a permutation. Let

$\pi$ be a permutation of length

$n$ . A raft of

$\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of

$\pi$ . See Definition 3.10 and Example 3.11 in [1].

Matching statistic: St001414

Mapping

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001414: Binary words ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%

Values

[[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => 111000 => 1 = 0 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 1 = 0 + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 1 = 0 + 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 2 = 1 + 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 2 = 1 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 2 = 1 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 1 = 0 + 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 1 = 0 + 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 1 = 0 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 1 = 0 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 1 = 0 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 1 = 0 + 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 1 = 0 + 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 1 = 0 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 1 = 0 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 1 = 0 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 1 = 0 + 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 1 = 0 + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 1 = 0 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 0 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? = 0 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 1 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => ? = 1 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 0 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 0 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? = 0 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 0 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 0 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? = 0 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 0 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => ? = 1 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 0 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => ? = 0 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 1 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 1 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? = 0 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? = 0 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => ? = 0 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 1 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 1 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? = 0 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? = 0 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => ? = 0 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 1 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 1 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 1 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? = 0 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 1 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 1 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => ? = 1 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? = 2 + 1

Description

Half the length of the longest odd length palindromic prefix of a binary word. More precisely, this statistic is the largest number

$k$ such that the word has a palindromic prefix of length

$2k+1$ .