- FindStat

Identifier

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001007: Dyck paths ⟶ ℤ (values match St000024The number of double up and double down steps of a Dyck path., St000443The number of long tunnels of a 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] => [1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 268 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

search for individual values

searching the database for the individual values of this statistic

Description

Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

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 connected components

Description

Sends a graph to the partition recording the number of edges in its connected components.

Map

Delest-Viennot

Description

Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
The Delest-Viennot bijection

$\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path

$(\gamma^{(-1)}\circ\beta)(D)$ .