- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000455: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 131 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{2} = 2$

Description

The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.

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

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

to graph

Description

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