Identifier
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Images
=>
Cc0005;cc-rep-1Cc0009;cc-rep-2
[1]=>[1,0]=>{{1}}=>[1]
[1,1]=>[1,0,1,0]=>{{1},{2}}=>[1,1]
[2]=>[1,1,0,0]=>{{1,2}}=>[2]
[1,1,1]=>[1,0,1,0,1,0]=>{{1},{2},{3}}=>[1,1,1]
[1,2]=>[1,0,1,1,0,0]=>{{1},{2,3}}=>[1,2]
[2,1]=>[1,1,0,0,1,0]=>{{1,2},{3}}=>[2,1]
[3]=>[1,1,1,0,0,0]=>{{1,2,3}}=>[3]
[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4}}=>[1,1,1,1]
[1,1,2]=>[1,0,1,0,1,1,0,0]=>{{1},{2},{3,4}}=>[1,1,2]
[1,2,1]=>[1,0,1,1,0,0,1,0]=>{{1},{2,3},{4}}=>[1,2,1]
[1,3]=>[1,0,1,1,1,0,0,0]=>{{1},{2,3,4}}=>[1,3]
[2,1,1]=>[1,1,0,0,1,0,1,0]=>{{1,2},{3},{4}}=>[2,1,1]
[2,2]=>[1,1,0,0,1,1,0,0]=>{{1,2},{3,4}}=>[2,2]
[3,1]=>[1,1,1,0,0,0,1,0]=>{{1,2,3},{4}}=>[3,1]
[4]=>[1,1,1,1,0,0,0,0]=>{{1,2,3,4}}=>[4]
[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5}}=>[1,1,1,1,1]
[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4,5}}=>[1,1,1,2]
[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>{{1},{2},{3,4},{5}}=>[1,1,2,1]
[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3,4,5}}=>[1,1,3]
[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>{{1},{2,3},{4},{5}}=>[1,2,1,1]
[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>{{1},{2,3},{4,5}}=>[1,2,2]
[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>{{1},{2,3,4},{5}}=>[1,3,1]
[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>{{1},{2,3,4,5}}=>[1,4]
[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5}}=>[2,1,1,1]
[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>{{1,2},{3},{4,5}}=>[2,1,2]
[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>{{1,2},{3,4},{5}}=>[2,2,1]
[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>{{1,2},{3,4,5}}=>[2,3]
[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>{{1,2,3},{4},{5}}=>[3,1,1]
[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>{{1,2,3},{4,5}}=>[3,2]
[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>{{1,2,3,4},{5}}=>[4,1]
[5]=>[1,1,1,1,1,0,0,0,0,0]=>{{1,2,3,4,5}}=>[5]
[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6}}=>[1,1,1,1,1,1]
[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5,6}}=>[1,1,1,1,2]
[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>{{1},{2},{3},{4,5},{6}}=>[1,1,1,2,1]
[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3},{4,5,6}}=>[1,1,1,3]
[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>{{1},{2},{3,4},{5},{6}}=>[1,1,2,1,1]
[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3,4},{5,6}}=>[1,1,2,2]
[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>{{1},{2},{3,4,5},{6}}=>[1,1,3,1]
[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3,4,5,6}}=>[1,1,4]
[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>{{1},{2,3},{4},{5},{6}}=>[1,2,1,1,1]
[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>{{1},{2,3},{4},{5,6}}=>[1,2,1,2]
[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>{{1},{2,3},{4,5},{6}}=>[1,2,2,1]
[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2,3},{4,5,6}}=>[1,2,3]
[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>{{1},{2,3,4},{5},{6}}=>[1,3,1,1]
[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>{{1},{2,3,4},{5,6}}=>[1,3,2]
[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>{{1},{2,3,4,5},{6}}=>[1,4,1]
[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2,3,4,5,6}}=>[1,5]
[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6}}=>[2,1,1,1,1]
[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5,6}}=>[2,1,1,2]
[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>{{1,2},{3},{4,5},{6}}=>[2,1,2,1]
[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>{{1,2},{3},{4,5,6}}=>[2,1,3]
[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>{{1,2},{3,4},{5},{6}}=>[2,2,1,1]
[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>{{1,2},{3,4},{5,6}}=>[2,2,2]
[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>{{1,2},{3,4,5},{6}}=>[2,3,1]
[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>{{1,2},{3,4,5,6}}=>[2,4]
[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6}}=>[3,1,1,1]
[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>{{1,2,3},{4},{5,6}}=>[3,1,2]
[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>{{1,2,3},{4,5},{6}}=>[3,2,1]
[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>{{1,2,3},{4,5,6}}=>[3,3]
[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>{{1,2,3,4},{5},{6}}=>[4,1,1]
[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>{{1,2,3,4},{5,6}}=>[4,2]
[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>{{1,2,3,4,5},{6}}=>[5,1]
[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>{{1,2,3,4,5,6}}=>[6]
[1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7}}=>[1,1,1,1,1,1,1]
[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6,7}}=>[1,1,1,1,1,2]
[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>{{1},{2},{3},{4},{5,6},{7}}=>[1,1,1,1,2,1]
[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3},{4},{5,6,7}}=>[1,1,1,1,3]
[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>{{1},{2},{3},{4,5},{6},{7}}=>[1,1,1,2,1,1]
[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3},{4,5},{6,7}}=>[1,1,1,2,2]
[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>{{1},{2},{3},{4,5,6},{7}}=>[1,1,1,3,1]
[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3},{4,5,6,7}}=>[1,1,1,4]
[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>{{1},{2},{3,4},{5},{6},{7}}=>[1,1,2,1,1,1]
[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>{{1},{2},{3,4},{5},{6,7}}=>[1,1,2,1,2]
[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>{{1},{2},{3,4},{5,6},{7}}=>[1,1,2,2,1]
[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2},{3,4},{5,6,7}}=>[1,1,2,3]
[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>{{1},{2},{3,4,5},{6},{7}}=>[1,1,3,1,1]
[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>{{1},{2},{3,4,5},{6,7}}=>[1,1,3,2]
[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>{{1},{2},{3,4,5,6},{7}}=>[1,1,4,1]
[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2},{3,4,5,6,7}}=>[1,1,5]
[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1},{2,3},{4},{5},{6},{7}}=>[1,2,1,1,1,1]
[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>{{1},{2,3},{4},{5},{6,7}}=>[1,2,1,1,2]
[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>{{1},{2,3},{4},{5,6},{7}}=>[1,2,1,2,1]
[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>{{1},{2,3},{4},{5,6,7}}=>[1,2,1,3]
[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>{{1},{2,3},{4,5},{6},{7}}=>[1,2,2,1,1]
[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1},{2,3},{4,5},{6,7}}=>[1,2,2,2]
[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>{{1},{2,3},{4,5,6},{7}}=>[1,2,3,1]
[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1},{2,3},{4,5,6,7}}=>[1,2,4]
[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>{{1},{2,3,4},{5},{6},{7}}=>[1,3,1,1,1]
[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>{{1},{2,3,4},{5},{6,7}}=>[1,3,1,2]
[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>{{1},{2,3,4},{5,6},{7}}=>[1,3,2,1]
[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1},{2,3,4},{5,6,7}}=>[1,3,3]
[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>{{1},{2,3,4,5},{6},{7}}=>[1,4,1,1]
[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>{{1},{2,3,4,5},{6,7}}=>[1,4,2]
[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>{{1},{2,3,4,5,6},{7}}=>[1,5,1]
[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7}}=>[1,6]
[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6},{7}}=>[2,1,1,1,1,1]
[2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5},{6,7}}=>[2,1,1,1,2]
[2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>{{1,2},{3},{4},{5,6},{7}}=>[2,1,1,2,1]
[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>{{1,2},{3},{4},{5,6,7}}=>[2,1,1,3]
[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>{{1,2},{3},{4,5},{6},{7}}=>[2,1,2,1,1]
[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>{{1,2},{3},{4,5},{6,7}}=>[2,1,2,2]
[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>{{1,2},{3},{4,5,6},{7}}=>[2,1,3,1]
[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>{{1,2},{3},{4,5,6,7}}=>[2,1,4]
[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5},{6},{7}}=>[2,2,1,1,1]
[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>{{1,2},{3,4},{5},{6,7}}=>[2,2,1,2]
[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1,2},{3,4},{5,6},{7}}=>[2,2,2,1]
[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>{{1,2},{3,4},{5,6,7}}=>[2,2,3]
[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>{{1,2},{3,4,5},{6},{7}}=>[2,3,1,1]
[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>{{1,2},{3,4,5},{6,7}}=>[2,3,2]
[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>{{1,2},{3,4,5,6},{7}}=>[2,4,1]
[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2},{3,4,5,6,7}}=>[2,5]
[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6},{7}}=>[3,1,1,1,1]
[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>{{1,2,3},{4},{5},{6,7}}=>[3,1,1,2]
[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>{{1,2,3},{4},{5,6},{7}}=>[3,1,2,1]
[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>{{1,2,3},{4},{5,6,7}}=>[3,1,3]
[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>{{1,2,3},{4,5},{6},{7}}=>[3,2,1,1]
[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>{{1,2,3},{4,5},{6,7}}=>[3,2,2]
[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>{{1,2,3},{4,5,6},{7}}=>[3,3,1]
[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3},{4,5,6,7}}=>[3,4]
[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>{{1,2,3,4},{5},{6},{7}}=>[4,1,1,1]
[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>{{1,2,3,4},{5},{6,7}}=>[4,1,2]
[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>{{1,2,3,4},{5,6},{7}}=>[4,2,1]
[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>{{1,2,3,4},{5,6,7}}=>[4,3]
[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>{{1,2,3,4,5},{6},{7}}=>[5,1,1]
[5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>{{1,2,3,4,5},{6,7}}=>[5,2]
[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6},{7}}=>[6,1]
[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1,2,3,4,5,6,7}}=>[7]
[1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7},{8}}=>[1,1,1,1,1,1,1,1]
[1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6},{7,8}}=>[1,1,1,1,1,1,2]
[1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3},{4},{5},{6,7,8}}=>[1,1,1,1,1,3]
[1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>{{1},{2},{3},{4},{5,6},{7},{8}}=>[1,1,1,1,2,1,1]
[1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3},{4},{5,6},{7,8}}=>[1,1,1,1,2,2]
[1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>{{1},{2},{3},{4},{5,6,7},{8}}=>[1,1,1,1,3,1]
[1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3},{4},{5,6,7,8}}=>[1,1,1,1,4]
[1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>{{1},{2},{3},{4,5},{6},{7,8}}=>[1,1,1,2,1,2]
[1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2},{3},{4,5,6,7,8}}=>[1,1,1,5]
[1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3,4},{5},{6},{7,8}}=>[1,1,2,1,1,2]
[1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>{{1},{2},{3,4},{5,6},{7},{8}}=>[1,1,2,2,1,1]
[1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3,4},{5,6,7,8}}=>[1,1,2,4]
[1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>{{1},{2},{3,4,5},{6},{7},{8}}=>[1,1,3,1,1,1]
[1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1},{2},{3,4,5},{6,7,8}}=>[1,1,3,3]
[1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>{{1},{2},{3,4,5,6},{7,8}}=>[1,1,4,2]
[1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>{{1},{2},{3,4,5,6,7},{8}}=>[1,1,5,1]
[1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1},{2},{3,4,5,6,7,8}}=>[1,1,6]
[1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2,3},{4},{5},{6},{7,8}}=>[1,2,1,1,1,2]
[1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>{{1},{2,3},{4},{5},{6,7},{8}}=>[1,2,1,1,2,1]
[1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2,3},{4},{5,6,7,8}}=>[1,2,1,4]
[1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1},{2,3},{4,5},{6,7},{8}}=>[1,2,2,2,1]
[1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2,3},{4,5},{6,7,8}}=>[1,2,2,3]
[1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>{{1},{2,3},{4,5,6},{7,8}}=>[1,2,3,2]
[1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2,3},{4,5,6,7,8}}=>[1,2,5]
[1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>{{1},{2,3,4},{5},{6},{7},{8}}=>[1,3,1,1,1,1]
[1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>{{1},{2,3,4},{5},{6,7,8}}=>[1,3,1,3]
[1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>{{1},{2,3,4},{5,6},{7,8}}=>[1,3,2,2]
[1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>{{1},{2,3,4},{5,6,7},{8}}=>[1,3,3,1]
[1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>{{1},{2,3,4,5,6},{7},{8}}=>[1,5,1,1]
[1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>{{1},{2,3,4,5,6},{7,8}}=>[1,5,2]
[1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>{{1},{2,3,4,5,6,7},{8}}=>[1,6,1]
[1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7,8}}=>[1,7]
[2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6},{7},{8}}=>[2,1,1,1,1,1,1]
[2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5},{6},{7,8}}=>[2,1,1,1,1,2]
[2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>{{1,2},{3},{4},{5},{6,7},{8}}=>[2,1,1,1,2,1]
[2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>{{1,2},{3},{4},{5},{6,7,8}}=>[2,1,1,1,3]
[2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>{{1,2},{3},{4},{5,6},{7},{8}}=>[2,1,1,2,1,1]
[2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>{{1,2},{3},{4},{5,6,7,8}}=>[2,1,1,4]
[2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>{{1,2},{3},{4,5},{6},{7},{8}}=>[2,1,2,1,1,1]
[2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>{{1,2},{3},{4,5},{6,7,8}}=>[2,1,2,3]
[2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>{{1,2},{3},{4,5,6},{7,8}}=>[2,1,3,2]
[2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2},{3},{4,5,6,7,8}}=>[2,1,5]
[2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5},{6},{7},{8}}=>[2,2,1,1,1,1]
[2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>{{1,2},{3,4},{5},{6,7,8}}=>[2,2,1,3]
[2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1,2},{3,4},{5,6},{7,8}}=>[2,2,2,2]
[2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1,2},{3,4},{5,6,7,8}}=>[2,2,4]
[2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>{{1,2},{3,4,5},{6},{7},{8}}=>[2,3,1,1,1]
[2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>{{1,2},{3,4,5,6},{7,8}}=>[2,4,2]
[2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>{{1,2},{3,4,5,6,7},{8}}=>[2,5,1]
[2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1,2},{3,4,5,6,7,8}}=>[2,6]
[3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6},{7},{8}}=>[3,1,1,1,1,1]
[3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2,3},{4},{5},{6},{7,8}}=>[3,1,1,1,2]
[3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>{{1,2,3},{4},{5},{6,7,8}}=>[3,1,1,3]
[3,1,2,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]=>{{1,2,3},{4},{5,6},{7,8}}=>[3,1,2,2]
[3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>{{1,2,3},{4,5,6,7},{8}}=>[3,4,1]
[3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2,3},{4,5,6,7,8}}=>[3,5]
[4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4},{5},{6},{7},{8}}=>[4,1,1,1,1]
[4,1,2,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]=>{{1,2,3,4},{5},{6,7},{8}}=>[4,1,2,1]
[4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>{{1,2,3,4},{5,6},{7,8}}=>[4,2,2]
[4,3,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>{{1,2,3,4},{5,6,7},{8}}=>[4,3,1]
[4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3,4},{5,6,7,8}}=>[4,4]
[5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>{{1,2,3,4,5},{6},{7},{8}}=>[5,1,1,1]
[5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>{{1,2,3,4,5},{6,7},{8}}=>[5,2,1]
[5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>{{1,2,3,4,5},{6,7,8}}=>[5,3]
[6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>{{1,2,3,4,5,6},{7},{8}}=>[6,1,1]
[6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>{{1,2,3,4,5,6},{7,8}}=>[6,2]
[7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6,7},{8}}=>[7,1]
[8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>{{1,2,3,4,5,6,7,8}}=>[8]
[1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9}}=>[1,1,1,1,1,1,1,1,1]
[1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6},{7},{8,9}}=>[1,1,1,1,1,1,1,2]
[1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3},{4},{5},{6},{7,8,9}}=>[1,1,1,1,1,1,3]
[1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6,7},{8,9}}=>[1,1,1,1,1,2,2]
[1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4,5},{6},{7},{8},{9}}=>[1,1,1,2,1,1,1,1]
[1,1,2,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3,4},{5},{6},{7},{8},{9}}=>[1,1,2,1,1,1,1,1]
[1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1},{2},{3,4,5,6,7,8,9}}=>[1,1,7]
[1,2,1,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2,3},{4},{5},{6},{7},{8,9}}=>[1,2,1,1,1,1,2]
[1,2,2,1,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1},{2,3},{4,5},{6},{7},{8},{9}}=>[1,2,2,1,1,1,1]
[1,3,1,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2,3,4},{5},{6},{7},{8},{9}}=>[1,3,1,1,1,1,1]
[1,6,2]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>{{1},{2,3,4,5,6,7},{8,9}}=>[1,6,2]
[1,7,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>{{1},{2,3,4,5,6,7,8},{9}}=>[1,7,1]
[1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7,8,9}}=>[1,8]
[2,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6},{7},{8},{9}}=>[2,1,1,1,1,1,1,1]
[2,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5},{6},{7},{8,9}}=>[2,1,1,1,1,1,2]
[2,1,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>{{1,2},{3},{4},{5},{6},{7,8},{9}}=>[2,1,1,1,1,2,1]
[2,1,2,1,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4,5},{6},{7},{8},{9}}=>[2,1,2,1,1,1,1]
[2,2,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5},{6},{7},{8},{9}}=>[2,2,1,1,1,1,1]
[2,6,1]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>{{1,2},{3,4,5,6,7,8},{9}}=>[2,6,1]
[2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1,2},{3,4,5,6,7,8,9}}=>[2,7]
[3,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6},{7},{8},{9}}=>[3,1,1,1,1,1,1]
[3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1,2,3},{4,5,6,7,8,9}}=>[3,6]
[4,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4},{5},{6},{7},{8},{9}}=>[4,1,1,1,1,1]
[4,5]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2,3,4},{5,6,7,8,9}}=>[4,5]
[5,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4,5},{6},{7},{8},{9}}=>[5,1,1,1,1]
[5,3,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]=>{{1,2,3,4,5},{6,7,8},{9}}=>[5,3,1]
[5,4]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3,4,5},{6,7,8,9}}=>[5,4]
[6,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]=>{{1,2,3,4,5,6},{7},{8},{9}}=>[6,1,1,1]
[6,2,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]=>{{1,2,3,4,5,6},{7,8},{9}}=>[6,2,1]
[6,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]=>{{1,2,3,4,5,6},{7,8,9}}=>[6,3]
[7,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>{{1,2,3,4,5,6,7},{8},{9}}=>[7,1,1]
[7,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>{{1,2,3,4,5,6,7},{8,9}}=>[7,2]
[8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6,7,8},{9}}=>[8,1]
[9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>{{1,2,3,4,5,6,7,8,9}}=>[9]
[1,1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}=>[1,1,1,1,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}=>[1,1,1,1,1,1,1,1,2]
[1,1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6,7},{8},{9},{10}}=>[1,1,1,1,1,2,1,1,1]
[1,8,1]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>{{1},{2,3,4,5,6,7,8,9},{10}}=>[1,8,1]
[1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7,8,9,10}}=>[1,9]
[2,1,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}=>[2,1,1,1,1,1,1,1,1]
[2,1,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5},{6},{7},{8},{9,10}}=>[2,1,1,1,1,1,1,2]
[2,2,1,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5},{6},{7},{8},{9},{10}}=>[2,2,1,1,1,1,1,1]
[2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>{{1,2},{3,4,5,6,7,8,9,10}}=>[2,8]
[3,1,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6},{7},{8},{9},{10}}=>[3,1,1,1,1,1,1,1]
[3,7]=>[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1,2,3},{4,5,6,7,8,9,10}}=>[3,7]
[4,1,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4},{5},{6},{7},{8},{9},{10}}=>[4,1,1,1,1,1,1]
[4,6]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1,2,3,4},{5,6,7,8,9,10}}=>[4,6]
[5,1,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4,5},{6},{7},{8},{9},{10}}=>[5,1,1,1,1,1]
[5,5]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2,3,4,5},{6,7,8,9,10}}=>[5,5]
[6,1,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4,5,6},{7},{8},{9},{10}}=>[6,1,1,1,1]
[6,4]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3,4,5,6},{7,8,9,10}}=>[6,4]
[7,1,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]=>{{1,2,3,4,5,6,7},{8},{9},{10}}=>[7,1,1,1]
[7,2,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]=>{{1,2,3,4,5,6,7},{8,9},{10}}=>[7,2,1]
[7,3]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]=>{{1,2,3,4,5,6,7},{8,9,10}}=>[7,3]
[8,1,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]=>{{1,2,3,4,5,6,7,8},{9},{10}}=>[8,1,1]
[8,2]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>{{1,2,3,4,5,6,7,8},{9,10}}=>[8,2]
[9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6,7,8,9},{10}}=>[9,1]
[10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>{{1,2,3,4,5,6,7,8,9,10}}=>[10]
[1,10]=>[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7,8,9,10,11}}=>[1,10]
[10,1]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6,7,8,9,10},{11}}=>[10,1]
[1,1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}=>[1,1,1,1,1,1,1,1,1,2]
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to noncrossing partition
Description
Biane's map to noncrossing set partitions.
Map
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
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