- FindStat

Identifier

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

([],5) => [5] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 274 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => [4,1,1,1] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => 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)],7) => [1,3,1,1,1] => [3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => 7

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

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

([(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) => [5,1,1] => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7

([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => [1,6,1] => [6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => 9

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

([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9) => [1,3,3,1,1] => [3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => 7

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

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

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

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

([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

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

([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [3,1,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,1,1,3,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => 9

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

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

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

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

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

([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

([(0,7),(1,2),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [3,1,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,1),(0,4),(0,5),(0,6),(0,7),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [3,1,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [3,1,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8) => [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

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

([(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

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

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

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

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

([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(6,7)],8) => [1,1,1,1,3,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => [1,6,1] => [6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => 9

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

([(0,3),(0,4),(0,7),(1,2),(1,5),(1,6),(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,5),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,7),(3,6),(4,6),(5,6),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

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

([(0,1),(0,7),(1,6),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

([(3,8),(4,7),(5,6),(6,8),(7,8)],9) => [1,1,1,1,1,4] => [4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 8

([(4,9),(5,8),(6,7),(7,9),(8,9)],10) => [1,1,1,1,1,5] => [5,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 7

([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9) => [1,1,1,1,1,4] => [4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 8

([(5,10),(6,9),(7,8),(8,10),(9,10)],11) => [1,1,1,1,1,6] => [6,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6

([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,7),(5,6),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,4),(0,7),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,5),(3,6),(4,7),(5,6),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,5),(0,6),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

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

([(0,1),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,4),(0,7),(1,2),(1,5),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,6),(5,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(5,6),(5,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,5),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,5),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(4,6),(5,6),(6,7)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,2),(0,7),(1,4),(1,5),(1,6),(2,3),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,4),(1,2),(1,5),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,6),(5,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,2),(1,2),(1,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,3),(0,4),(0,6),(1,2),(1,5),(1,7),(2,5),(2,7),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,7),(1,7),(2,7),(3,4),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,4),(0,5),(1,2),(1,3),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,2),(1,2),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

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

([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,6),(1,4),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,7),(1,4),(1,7),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,4),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(5,6),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(5,6),(5,7),(6,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(3,4),(3,7),(4,7),(5,6),(5,7)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6)],8) => [1,1,1,3,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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

([(0,4),(0,5),(1,4),(1,5),(2,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,3),(0,4),(0,7),(1,2),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8) => [1,1,3,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

([(0,2),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9

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$F_{1} = q$

$F_{2} = q + q^{2}$

Description

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

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

Laplacian multiplicities

Description

The composition of multiplicities of the Laplacian eigenvalues.
Let

$\lambda_1 > \lambda_2 > \dots$ be the eigenvalues of the Laplacian matrix of a graph on

$n$ vertices. Then this map returns the composition

$a_1,\dots,a_k$ of

$n$ where

$a_i$ is the multiplicity of

$\lambda_i$ .

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.