- FindStat

Identifier

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤ

Values

[[1]] => [1,0] => [1] => ([],1) => 0

[[1,0],[0,1]] => [1,0,1,0] => [1,2] => ([],2) => 0

[[0,1],[1,0]] => [1,1,0,0] => [2,1] => ([(0,1)],2) => 1

[[1,0,0],[0,1,0],[0,0,1]] => [1,0,1,0,1,0] => [1,2,3] => ([],3) => 0

[[0,1,0],[1,0,0],[0,0,1]] => [1,1,0,0,1,0] => [2,1,3] => ([(1,2)],3) => 1

[[1,0,0],[0,0,1],[0,1,0]] => [1,0,1,1,0,0] => [1,3,2] => ([(1,2)],3) => 1

[[0,0,1],[1,0,0],[0,1,0]] => [1,1,1,0,0,0] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2

[[0,0,1],[0,1,0],[1,0,0]] => [1,1,1,0,0,0] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

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

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

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

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

[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

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

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

[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

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

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

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

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

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

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

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

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

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

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

[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

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

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

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

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

[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

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

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

[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

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

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

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

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

[[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [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) => 4

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

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

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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.

Map

to non-crossing permutation

Description

Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Map

graph of inversions

Description

The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.

Map

to Dyck path

Description

The Dyck path determined by the last diagonal of the monotone triangle of an alternating sign matrix.