Identifier
  • Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
    Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
    St000443: Dyck paths ⟶ ℤ (values match St000024The number of double up and double down steps of a Dyck path., St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path., St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra., St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.)
Values
([(0,1)],2) => [1] => [1,0] => 1
([(1,2)],3) => [1] => [1,0] => 1
([(0,2),(1,2)],3) => [1,1] => [1,1,0,0] => 2
([(0,1),(0,2),(1,2)],3) => [3] => [1,0,1,0,1,0] => 1
([(2,3)],4) => [1] => [1,0] => 1
([(1,3),(2,3)],4) => [1,1] => [1,1,0,0] => 2
([(0,3),(1,3),(2,3)],4) => [1,1,1] => [1,1,0,1,0,0] => 2
([(0,3),(1,2)],4) => [1,1] => [1,1,0,0] => 2
([(0,3),(1,2),(2,3)],4) => [1,1,1] => [1,1,0,1,0,0] => 2
([(1,2),(1,3),(2,3)],4) => [3] => [1,0,1,0,1,0] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => 2
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,0,1,0,1,0,1,0] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,0,1,0,1,0,1,0,1,0] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(3,4)],5) => [1] => [1,0] => 1
([(2,4),(3,4)],5) => [1,1] => [1,1,0,0] => 2
([(1,4),(2,4),(3,4)],5) => [1,1,1] => [1,1,0,1,0,0] => 2
([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(1,4),(2,3)],5) => [1,1] => [1,1,0,0] => 2
([(1,4),(2,3),(3,4)],5) => [1,1,1] => [1,1,0,1,0,0] => 2
([(0,1),(2,4),(3,4)],5) => [1,1,1] => [1,1,0,1,0,0] => 2
([(2,3),(2,4),(3,4)],5) => [3] => [1,0,1,0,1,0] => 1
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => [1,0,1,0,1,1,0,0] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,0,1,0,1,0,1,0] => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,0,1,0,1,1,0,0] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,1,0,0,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(4,5)],6) => [1] => [1,0] => 1
([(3,5),(4,5)],6) => [1,1] => [1,1,0,0] => 2
([(2,5),(3,5),(4,5)],6) => [1,1,1] => [1,1,0,1,0,0] => 2
([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(2,5),(3,4)],6) => [1,1] => [1,1,0,0] => 2
([(2,5),(3,4),(4,5)],6) => [1,1,1] => [1,1,0,1,0,0] => 2
([(1,2),(3,5),(4,5)],6) => [1,1,1] => [1,1,0,1,0,0] => 2
([(3,4),(3,5),(4,5)],6) => [3] => [1,0,1,0,1,0] => 1
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,0,1,0,1,0,1,0] => 1
([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,1,0,1,0,0] => 2
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3] => [1,1,1,0,1,0,0,0] => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,1,0,0,0] => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
([(5,6)],7) => [1] => [1,0] => 1
([(4,6),(5,6)],7) => [1,1] => [1,1,0,0] => 2
([(3,6),(4,6),(5,6)],7) => [1,1,1] => [1,1,0,1,0,0] => 2
>>> Load all 212 entries. <<<
([(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(3,6),(4,5)],7) => [1,1] => [1,1,0,0] => 2
([(3,6),(4,5),(5,6)],7) => [1,1,1] => [1,1,0,1,0,0] => 2
([(2,3),(4,6),(5,6)],7) => [1,1,1] => [1,1,0,1,0,0] => 2
([(4,5),(4,6),(5,6)],7) => [3] => [1,0,1,0,1,0] => 1
([(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(1,2),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(3,6),(4,5),(4,6),(5,6)],7) => [3,1] => [1,0,1,0,1,1,0,0] => 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(3,5),(3,6),(4,5),(4,6)],7) => [4] => [1,0,1,0,1,0,1,0] => 1
([(1,6),(2,6),(3,5),(4,5)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(1,6),(2,5),(3,4)],7) => [1,1,1] => [1,1,0,1,0,0] => 2
([(2,6),(3,5),(4,5),(4,6)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(1,2),(3,6),(4,5),(5,6)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(0,3),(1,2),(4,6),(5,6)],7) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
([(2,3),(4,5),(4,6),(5,6)],7) => [3,1] => [1,0,1,0,1,1,0,0] => 2
([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3] => [1,1,1,0,1,0,0,0] => 3
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(1,6),(2,5),(3,4),(3,5),(4,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 3
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 3
([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(0,1),(0,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3
([(0,1),(2,5),(3,4),(4,6),(5,6)],7) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
([(0,5),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 5
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
([(0,4),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(5,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(0,1),(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3] => [1,1,1,0,1,0,0,0] => 3
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 5
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,6),(4,6),(5,6)],7) => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 3
([(0,6),(1,2),(1,4),(2,4),(3,5),(3,6),(4,5),(5,6)],7) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
([(0,1),(0,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
([(0,3),(0,6),(1,3),(1,6),(2,4),(2,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3
search for individual values
searching the database for the individual values of this statistic
Description
The number of long tunnels of a Dyck path.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to edge-partition of biconnected components
Description
Sends a graph to the partition recording the number of edges in its biconnected components.
The biconnected components are also known as blocks of a graph.