Identifier
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Images
=>
Cc0021;cc-rep-0Cc0020;cc-rep-1
[]=>([],1)=>[1]=>[1] [[]]=>([(0,1)],2)=>[1,1]=>[1,1] [[],[]]=>([(0,2),(1,2)],3)=>[2,1]=>[2,1] [[[]]]=>([(0,2),(1,2)],3)=>[2,1]=>[2,1] [[],[],[]]=>([(0,3),(1,3),(2,3)],4)=>[3,1]=>[3,1] [[],[[]]]=>([(0,3),(1,2),(2,3)],4)=>[2,2]=>[2,2] [[[]],[]]=>([(0,3),(1,2),(2,3)],4)=>[2,2]=>[2,2] [[[],[]]]=>([(0,3),(1,3),(2,3)],4)=>[3,1]=>[3,1] [[[[]]]]=>([(0,3),(1,2),(2,3)],4)=>[2,2]=>[2,2] [[],[],[],[]]=>([(0,4),(1,4),(2,4),(3,4)],5)=>[4,1]=>[4,1] [[],[],[[]]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,2]=>[3,2] [[],[[]],[]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,2]=>[3,2] [[],[[],[]]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,2]=>[3,2] [[],[[[]]]]=>([(0,4),(1,3),(2,3),(2,4)],5)=>[3,2]=>[3,2] [[[]],[],[]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,2]=>[3,2] [[[]],[[]]]=>([(0,4),(1,3),(2,3),(2,4)],5)=>[3,2]=>[3,2] [[[],[]],[]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,2]=>[3,2] [[[[]]],[]]=>([(0,4),(1,3),(2,3),(2,4)],5)=>[3,2]=>[3,2] [[[],[],[]]]=>([(0,4),(1,4),(2,4),(3,4)],5)=>[4,1]=>[4,1] [[[],[[]]]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,2]=>[3,2] [[[[]],[]]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,2]=>[3,2] [[[[],[]]]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,2]=>[3,2] [[[[[]]]]]=>([(0,4),(1,3),(2,3),(2,4)],5)=>[3,2]=>[3,2] [[],[],[],[],[]]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>[5,1]=>[5,1] [[],[],[],[[]]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[],[],[[]],[]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[],[],[[],[]]]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[],[],[[[]]]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[],[[]],[],[]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[],[[]],[[]]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[],[[],[]],[]]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[],[[[]]],[]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[],[[],[],[]]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[],[[],[[]]]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[],[[[]],[]]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[],[[[],[]]]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[],[[[[]]]]]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>[3,3]=>[3,3] [[[]],[],[],[]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[]],[],[[]]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[[]],[[]],[]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[[]],[[],[]]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[]],[[[]]]]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>[3,3]=>[3,3] [[[],[]],[],[]]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[[]]],[],[]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[],[]],[[]]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[[]]],[[]]]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>[3,3]=>[3,3] [[[],[],[]],[]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[],[[]]],[]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[[[]],[]],[]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[[[],[]]],[]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[[[]]]],[]]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>[3,3]=>[3,3] [[[],[],[],[]]]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>[5,1]=>[5,1] [[[],[],[[]]]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[],[[]],[]]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[],[[],[]]]]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[],[[[]]]]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[[]],[],[]]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[[]],[[]]]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[[[],[]],[]]]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[[[]]],[]]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[[],[],[]]]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[[],[[]]]]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[[[[]],[]]]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,3]=>[3,3] [[[[[],[]]]]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,2]=>[4,2] [[[[[[]]]]]]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>[3,3]=>[3,3] [[],[],[],[],[],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>[6,1]=>[6,1] [[],[],[],[],[[]]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[],[],[[]],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[],[],[[],[]]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[],[],[[[]]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[],[[]],[],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[],[[]],[[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[],[[],[]],[]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[],[[[]]],[]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[],[[],[],[]]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[],[[],[[]]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[],[[[]],[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[],[[[],[]]]]=>([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)=>[5,2]=>[5,2] [[],[],[[[[]]]]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[],[[]],[],[],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[[]],[],[[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[]],[[]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[]],[[],[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[]],[[[]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[],[[],[]],[],[]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[[[]]],[],[]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[[],[]],[[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[[]]],[[]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[],[[],[],[]],[]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[[],[[]]],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[[]],[]],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[[],[]]],[]]=>([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)=>[5,2]=>[5,2] [[],[[[[]]]],[]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[],[[],[],[],[]]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[[],[],[[]]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[],[[]],[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[],[[],[]]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[],[[[]]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[],[[[]],[],[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[[]],[[]]]]=>([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[[],[]],[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[],[[[[]]],[]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[],[[[],[],[]]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[],[[[],[[]]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[],[[[[]],[]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[],[[[[],[]]]]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[],[[[[[]]]]]]=>([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)=>[4,3]=>[4,3] [[[]],[],[],[],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[]],[],[],[[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[]],[],[[]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[]],[],[[],[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[]],[],[[[]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[]],[[]],[],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[]],[[]],[[]]]=>([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[]],[[],[]],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[]],[[[]]],[]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[]],[[],[],[]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[]],[[],[[]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[]],[[[]],[]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[]],[[[],[]]]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[[]],[[[[]]]]]=>([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)=>[4,3]=>[4,3] [[[],[]],[],[],[]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[[]]],[],[],[]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[]],[],[[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[]]],[],[[]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[],[]],[[]],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[]]],[[]],[]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[],[]],[[],[]]]=>([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)=>[5,2]=>[5,2] [[[],[]],[[[]]]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[[[]]],[[],[]]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[[[]]],[[[]]]]=>([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)=>[4,3]=>[4,3] [[[],[],[]],[],[]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[[]]],[],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[]],[]],[],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[],[]]],[],[]]=>([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)=>[5,2]=>[5,2] [[[[[]]]],[],[]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[[],[],[]],[[]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[[]]],[[]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[]],[]],[[]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[],[]]],[[]]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[[[[]]]],[[]]]=>([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)=>[4,3]=>[4,3] [[[],[],[],[]],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[],[[]]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[],[[]],[]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[],[[],[]]],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[],[[[]]]],[]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[]],[],[]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[]],[[]]],[]]=>([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[],[]],[]],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[[]]],[]],[]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[],[],[]]],[]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[[],[[]]]],[]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[[]],[]]],[]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[[],[]]]],[]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[[[[[]]]]],[]]=>([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)=>[4,3]=>[4,3] [[[],[],[],[],[]]]=>([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>[6,1]=>[6,1] [[[],[],[],[[]]]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[],[[]],[]]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[],[[],[]]]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[],[[[]]]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[[]],[],[]]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[[]],[[]]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[],[[],[]],[]]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[[[]]],[]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[[],[],[]]]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[],[[],[[]]]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[],[[[]],[]]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[],[[[],[]]]]]=>([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)=>[5,2]=>[5,2] [[[],[[[[]]]]]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[[[]],[],[],[]]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[[]],[],[[]]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[]],[[]],[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[]],[[],[]]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[]],[[[]]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[],[]],[],[]]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[[[]]],[],[]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[[],[]],[[]]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[[]]],[[]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[],[],[]],[]]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[[],[[]]],[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[[]],[]],[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[[],[]]],[]]]=>([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)=>[5,2]=>[5,2] [[[[[[]]]],[]]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[[[],[],[],[]]]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[[],[],[[]]]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[],[[]],[]]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[],[[],[]]]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[],[[[]]]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[[]],[],[]]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[[]],[[]]]]]=>([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[[],[]],[]]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[4,3]=>[4,3] [[[[[[]]],[]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[[],[],[]]]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>[5,2]=>[5,2] [[[[[],[[]]]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[[[]],[]]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>[4,3]=>[4,3] [[[[[[],[]]]]]]=>([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>[4,3]=>[4,3] [[[[[[[]]]]]]]=>([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)=>[4,3]=>[4,3]
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges.
Map
chromatic difference sequence
Description
The chromatic difference sequence of a graph.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00314Foata bijection.