Identifier
Values
([(0,1)],2) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(1,2)],3) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(0,2),(1,2)],3) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 0
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(2,3)],4) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(1,3),(2,3)],4) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 0
([(0,3),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(0,3),(1,2)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(0,3),(1,2),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(3,4)],5) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(2,4),(3,4)],5) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 0
([(1,4),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(0,4),(1,4),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(1,4),(2,3)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(1,4),(2,3),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(0,1),(2,4),(3,4)],5) => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(0,4),(1,4),(2,3),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(1,4),(2,3),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,4),(1,3),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(4,5)],6) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(3,5),(4,5)],6) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 0
([(2,5),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(1,5),(2,5),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(2,5),(3,4)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(2,5),(3,4),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(1,2),(3,5),(4,5)],6) => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(1,5),(2,5),(3,4),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,1),(2,5),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
([(2,5),(3,4),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 1
([(1,5),(2,4),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(0,1),(2,5),(3,4),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1
([(5,6)],7) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(4,6),(5,6)],7) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 0
([(3,6),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(2,6),(3,6),(4,6),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(3,6),(4,5)],7) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(3,6),(4,5),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(2,3),(4,6),(5,6)],7) => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0
([(4,5),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
([(2,6),(3,6),(4,5),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(1,2),(3,6),(4,6),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
([(3,6),(4,5),(4,6),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(3,5),(3,6),(4,5),(4,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(1,6),(2,6),(3,5),(4,5)],7) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,6),(2,5),(3,4)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 1
([(2,6),(3,5),(4,5),(4,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(1,2),(3,6),(4,5),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
([(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] => 1
([(2,3),(4,5),(4,6),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
([(1,6),(2,5),(3,4),(3,5),(4,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
>>> Load all 120 entries. <<<
([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7) => [5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 0
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 0
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 0
search for individual values
searching the database for the individual values of this statistic
Description
Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to edge-partition of connected components
Description
Sends a graph to the partition recording the number of edges in its connected components.