- FindStat

Identifier

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 208 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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] => [6] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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] => [6] => 1

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

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

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

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

([(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] => [6] => 1

([(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] => [6] => 1

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

([(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] => [6] => 1

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

([(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] => [6] => 1

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

([(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] => [6] => 1

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

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

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

([(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] => [6] => 1

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

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

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

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

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

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

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

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

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

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

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

([(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] => [6] => 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] => [6] => 1

([(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] => [6] => 1

([(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] => [6] => 1

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

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

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

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

([(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] => [6] => 1

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

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

([(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] => [6] => 1

([(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] => [6] => 1

([(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] => [6] => 1

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

([(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] => [6] => 1

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

([(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] => [6] => 1

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

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

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

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

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

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

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

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

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

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

([(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] => [1,1,1,3] => 4

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

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$F_{1} = q$

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

$F_{3} = q + 2\ q^{2} + q^{3}$

$F_{4} = 2\ q + 6\ q^{2} + 2\ q^{3} + q^{4}$

$F_{5} = 10\ q + 15\ q^{2} + 6\ q^{3} + 2\ q^{4} + q^{5}$

$F_{6} = 53\ q + 66\ q^{2} + 26\ q^{3} + 8\ q^{4} + 2\ q^{5} + q^{6}$

Description

The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

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

complement

Description

The complement of a composition.
The complement of a composition

$I$ is defined as follows:
If

$I$ is the empty composition, then the complement is also the empty composition. Otherwise, let

$S$ be the descent set corresponding to

$I=(i_1,\dots,i_k)$ , that is, the subset

$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$
of

$\{ 1, 2, \ldots, |I|-1 \}$ . Then, the complement of

$I$ is the composition of the same size as

$I$ , whose descent set is

$\{ 1, 2, \ldots, |I|-1 \} \setminus S$ .
The complement of a composition

$I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to

$I$ .