Identifier
-
Mp00152:
Graphs
—Laplacian multiplicities⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001198: Dyck paths ⟶ ℤ (values match St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.)
Values
([],2) => [2] => [1,1,0,0] => [1,0,1,0] => 2
([],3) => [3] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 2
([(1,2)],3) => [1,2] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 2
([(0,1),(0,2),(1,2)],3) => [2,1] => [1,1,0,0,1,0] => [1,1,0,1,0,0] => 2
([],4) => [4] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 2
([(2,3)],4) => [1,3] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 2
([(1,3),(2,3)],4) => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 2
([(0,3),(1,3),(2,3)],4) => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 2
([(0,3),(1,2)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => 2
([(1,2),(1,3),(2,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => 2
([(0,2),(0,3),(1,2),(1,3)],4) => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => 3
([],5) => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 2
([(3,4)],5) => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 2
([(2,4),(3,4)],5) => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 2
([(1,4),(2,4),(3,4)],5) => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => 2
([(0,4),(1,4),(2,4),(3,4)],5) => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(1,4),(2,3)],5) => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 2
([(1,4),(2,3),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2
([(0,1),(2,4),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2
([(2,3),(2,4),(3,4)],5) => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 2
([(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 2
([(1,3),(1,4),(2,3),(2,4)],5) => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => 3
([(0,1),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 2
([(4,5)],6) => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(3,5),(4,5)],6) => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 2
([(2,5),(3,5),(4,5)],6) => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 2
([(1,5),(2,5),(3,5),(4,5)],6) => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 3
([(2,5),(3,4)],6) => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(2,5),(3,4),(4,5)],6) => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
([(1,2),(3,5),(4,5)],6) => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
([(3,4),(3,5),(4,5)],6) => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
([(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 3
([(2,4),(2,5),(3,4),(3,5)],6) => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 2
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 3
([(0,5),(1,4),(2,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 3
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(0,1),(2,5),(3,4),(4,5)],6) => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
([(1,2),(3,4),(3,5),(4,5)],6) => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 3
([(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] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 3
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
>>> Load all 140 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Map
bounce path
Description
The bounce path determined by an integer composition.
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 $\lambda_i$.
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
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!