- FindStat

Identifier

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤ

Values

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

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$F_{1} = q^{2}$

$F_{2} = 2\ q^{3}$

Description

The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]

Map

to ordered tree

Description

Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path

$D$ of semilength

$n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths

$D_1,\dots,D_k$ of respective semilengths

$n_1,\dots,n_k$ (so one has

$n = n_1 + \dots n_k$ ) each of which has no returns.
Denote by

$\tilde D_i$ the path of semilength

$n_i-1$ obtained from

$D_i$ by removing the initial up- and the final down-step.
This map then sends

$D$ to the tree

$T$ having a root note with ordered children

$T_1,\dots,T_k$ which are again ordered trees computed from

$D_1,\dots,D_k$ respectively.
The unique path of semilength

$1$ is sent to the tree consisting of a single node.

Map

to Dyck path

Description

The Dyck path determined by the last diagonal of the monotone triangle of an alternating sign matrix.

Map

to graph

Description

Return the undirected graph obtained from the tree nodes and edges.