Identifier
Values
([(0,1)],2) => [1] => [1,0,1,0] => [1,0,1,0] => 0
([(1,2)],3) => [1] => [1,0,1,0] => [1,0,1,0] => 0
([(0,2),(1,2)],3) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(2,3)],4) => [1] => [1,0,1,0] => [1,0,1,0] => 0
([(1,3),(2,3)],4) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,3),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,3),(1,2)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(0,3),(1,2),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(3,4)],5) => [1] => [1,0,1,0] => [1,0,1,0] => 0
([(2,4),(3,4)],5) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(1,4),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,4),(1,4),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(1,4),(2,3)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(1,4),(2,3),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,1),(2,4),(3,4)],5) => [2,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => 0
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,4),(1,4),(2,3),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(1,4),(2,3),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,4),(1,3),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(4,5)],6) => [1] => [1,0,1,0] => [1,0,1,0] => 0
([(3,5),(4,5)],6) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(2,5),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(1,5),(2,5),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(2,5),(3,4)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(2,5),(3,4),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(1,2),(3,5),(4,5)],6) => [2,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => 0
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(1,5),(2,5),(3,4),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,1),(2,5),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 1
([(2,5),(3,4),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(1,5),(2,4),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,1),(2,5),(3,4),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 1
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 1
([(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] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
([(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] => [1,1,1,1,0,0,0,1,0,0] => 3
([(5,6)],7) => [1] => [1,0,1,0] => [1,0,1,0] => 0
([(4,6),(5,6)],7) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(3,6),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(2,6),(3,6),(4,6),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(3,6),(4,5)],7) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(3,6),(4,5),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(2,3),(4,6),(5,6)],7) => [2,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => 0
([(4,5),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(2,6),(3,6),(4,5),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(1,2),(3,6),(4,6),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 1
([(3,6),(4,5),(4,6),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(3,5),(3,6),(4,5),(4,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(1,6),(2,6),(3,5),(4,5)],7) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,6),(2,5),(3,4)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(2,6),(3,5),(4,5),(4,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(1,2),(3,6),(4,5),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 1
([(0,3),(1,2),(4,6),(5,6)],7) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => 2
([(2,3),(4,5),(4,6),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 1
([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 1
([(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] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,6),(2,5),(3,4),(3,5),(4,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
>>> 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] => [1,0,1,1,1,0,0,0,1,0] => 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 1
([(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] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
([(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] => 2
([(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] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
([(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] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
([(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] => [1,1,1,1,0,0,0,1,0,0] => 3
([(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] => 2
([(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] => [1,1,0,0,1,1,1,0,0,0,1,0] => 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 2
([(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] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
([(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] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
([(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] => [1,1,1,1,0,0,0,1,0,0] => 3
([(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] => [1,1,1,1,0,0,0,1,0,0] => 3
([(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] => 1
([(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] => 1
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0] => 2
search for individual values
searching the database for the individual values of this statistic
Description
The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary.
Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$.
This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly above the diagonal and relative to the diagonal boundary.
Map
switch returns and last double rise
Description
An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
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.