Identifier
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Images
=>
Cc0005;cc-rep-1Cc0009;cc-rep-3
[1]=>[1,0]=>[2,1]=>{{1,2}} [1,1]=>[1,0,1,0]=>[3,1,2]=>{{1,2,3}} [2]=>[1,1,0,0]=>[2,3,1]=>{{1,2,3}} [1,1,1]=>[1,0,1,0,1,0]=>[4,1,2,3]=>{{1,2,3,4}} [1,2]=>[1,0,1,1,0,0]=>[3,1,4,2]=>{{1,2,3,4}} [2,1]=>[1,1,0,0,1,0]=>[2,4,1,3]=>{{1,2,3,4}} [3]=>[1,1,1,0,0,0]=>[2,3,4,1]=>{{1,2,3,4}} [1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[5,1,2,3,4]=>{{1,2,3,4,5}} [1,1,2]=>[1,0,1,0,1,1,0,0]=>[4,1,2,5,3]=>{{1,2,3,4,5}} [1,2,1]=>[1,0,1,1,0,0,1,0]=>[3,1,5,2,4]=>{{1,2,3,4,5}} [1,3]=>[1,0,1,1,1,0,0,0]=>[3,1,4,5,2]=>{{1,2,3,4,5}} [2,1,1]=>[1,1,0,0,1,0,1,0]=>[2,5,1,3,4]=>{{1,2,3,4,5}} [2,2]=>[1,1,0,0,1,1,0,0]=>[2,4,1,5,3]=>{{1,2,3,4,5}} [3,1]=>[1,1,1,0,0,0,1,0]=>[2,3,5,1,4]=>{{1,2,3,4,5}} [4]=>[1,1,1,1,0,0,0,0]=>[2,3,4,5,1]=>{{1,2,3,4,5}} [1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[6,1,2,3,4,5]=>{{1,2,3,4,5,6}} [1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[5,1,2,3,6,4]=>{{1,2,3,4,5,6}} [1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[4,1,2,6,3,5]=>{{1,2,3,4,5,6}} [1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[4,1,2,5,6,3]=>{{1,2,3,4,5,6}} [1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[3,1,6,2,4,5]=>{{1,2,3,4,5,6}} [1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[3,1,5,2,6,4]=>{{1,2,3,4,5,6}} [1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[3,1,4,6,2,5]=>{{1,2,3,4,5,6}} [1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[3,1,4,5,6,2]=>{{1,2,3,4,5,6}} [2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[2,6,1,3,4,5]=>{{1,2,3,4,5,6}} [2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[2,5,1,3,6,4]=>{{1,2,3,4,5,6}} [2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[2,4,1,6,3,5]=>{{1,2,3,4,5,6}} [2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[2,4,1,5,6,3]=>{{1,2,3,4,5,6}} [3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[2,3,6,1,4,5]=>{{1,2,3,4,5,6}} [3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[2,3,5,1,6,4]=>{{1,2,3,4,5,6}} [4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[2,3,4,6,1,5]=>{{1,2,3,4,5,6}} [5]=>[1,1,1,1,1,0,0,0,0,0]=>[2,3,4,5,6,1]=>{{1,2,3,4,5,6}} [1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[7,1,2,3,4,5,6]=>{{1,2,3,4,5,6,7}} [1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[6,1,2,3,4,7,5]=>{{1,2,3,4,5,6,7}} [1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[5,1,2,3,7,4,6]=>{{1,2,3,4,5,6,7}} [1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[5,1,2,3,6,7,4]=>{{1,2,3,4,5,6,7}} [1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[4,1,2,7,3,5,6]=>{{1,2,3,4,5,6,7}} [1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[4,1,2,6,3,7,5]=>{{1,2,3,4,5,6,7}} [1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[4,1,2,5,7,3,6]=>{{1,2,3,4,5,6,7}} [1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[4,1,2,5,6,7,3]=>{{1,2,3,4,5,6,7}} [1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[3,1,7,2,4,5,6]=>{{1,2,3,4,5,6,7}} [1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[3,1,6,2,4,7,5]=>{{1,2,3,4,5,6,7}} [1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[3,1,5,2,7,4,6]=>{{1,2,3,4,5,6,7}} [1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[3,1,5,2,6,7,4]=>{{1,2,3,4,5,6,7}} [1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[3,1,4,7,2,5,6]=>{{1,2,3,4,5,6,7}} [1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[3,1,4,6,2,7,5]=>{{1,2,3,4,5,6,7}} [1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[3,1,4,5,7,2,6]=>{{1,2,3,4,5,6,7}} [1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[3,1,4,5,6,7,2]=>{{1,2,3,4,5,6,7}} [2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,7,1,3,4,5,6]=>{{1,2,3,4,5,6,7}} [2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,6,1,3,4,7,5]=>{{1,2,3,4,5,6,7}} [2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,5,1,3,7,4,6]=>{{1,2,3,4,5,6,7}} [2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,5,1,3,6,7,4]=>{{1,2,3,4,5,6,7}} [2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,4,1,7,3,5,6]=>{{1,2,3,4,5,6,7}} [2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,4,1,6,3,7,5]=>{{1,2,3,4,5,6,7}} [2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,4,1,5,7,3,6]=>{{1,2,3,4,5,6,7}} [2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,4,1,5,6,7,3]=>{{1,2,3,4,5,6,7}} [3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[2,3,7,1,4,5,6]=>{{1,2,3,4,5,6,7}} [3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[2,3,6,1,4,7,5]=>{{1,2,3,4,5,6,7}} [3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[2,3,5,1,7,4,6]=>{{1,2,3,4,5,6,7}} [3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[2,3,5,1,6,7,4]=>{{1,2,3,4,5,6,7}} [4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[2,3,4,7,1,5,6]=>{{1,2,3,4,5,6,7}} [4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[2,3,4,6,1,7,5]=>{{1,2,3,4,5,6,7}} [5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[2,3,4,5,7,1,6]=>{{1,2,3,4,5,6,7}} [6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,4,5,6,7,1]=>{{1,2,3,4,5,6,7}} [1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[8,1,2,3,4,5,6,7]=>{{1,2,3,4,5,6,7,8}} [1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[7,1,2,3,4,5,8,6]=>{{1,2,3,4,5,6,7,8}} [1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[6,1,2,3,4,8,5,7]=>{{1,2,3,4,5,6,7,8}} [1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[6,1,2,3,4,7,8,5]=>{{1,2,3,4,5,6,7,8}} [1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[5,1,2,3,8,4,6,7]=>{{1,2,3,4,5,6,7,8}} [1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[5,1,2,3,7,4,8,6]=>{{1,2,3,4,5,6,7,8}} [1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[5,1,2,3,6,8,4,7]=>{{1,2,3,4,5,6,7,8}} [1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[5,1,2,3,6,7,8,4]=>{{1,2,3,4,5,6,7,8}} [1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[4,1,2,8,3,5,6,7]=>{{1,2,3,4,5,6,7,8}} [1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[4,1,2,7,3,5,8,6]=>{{1,2,3,4,5,6,7,8}} [1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[4,1,2,6,3,8,5,7]=>{{1,2,3,4,5,6,7,8}} [1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[4,1,2,6,3,7,8,5]=>{{1,2,3,4,5,6,7,8}} [1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[4,1,2,5,8,3,6,7]=>{{1,2,3,4,5,6,7,8}} [1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[4,1,2,5,7,3,8,6]=>{{1,2,3,4,5,6,7,8}} [1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[4,1,2,5,6,8,3,7]=>{{1,2,3,4,5,6,7,8}} [1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[4,1,2,5,6,7,8,3]=>{{1,2,3,4,5,6,7,8}} [1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[3,1,8,2,4,5,6,7]=>{{1,2,3,4,5,6,7,8}} [1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[3,1,7,2,4,5,8,6]=>{{1,2,3,4,5,6,7,8}} [1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[3,1,6,2,4,8,5,7]=>{{1,2,3,4,5,6,7,8}} [1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[3,1,6,2,4,7,8,5]=>{{1,2,3,4,5,6,7,8}} [1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[3,1,5,2,8,4,6,7]=>{{1,2,3,4,5,6,7,8}} [1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[3,1,5,2,7,4,8,6]=>{{1,2,3,4,5,6,7,8}} [1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[3,1,5,2,6,8,4,7]=>{{1,2,3,4,5,6,7,8}} [1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[3,1,5,2,6,7,8,4]=>{{1,2,3,4,5,6,7,8}} [1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[3,1,4,8,2,5,6,7]=>{{1,2,3,4,5,6,7,8}} [1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[3,1,4,7,2,5,8,6]=>{{1,2,3,4,5,6,7,8}} [1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[3,1,4,6,2,8,5,7]=>{{1,2,3,4,5,6,7,8}} [1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[3,1,4,6,2,7,8,5]=>{{1,2,3,4,5,6,7,8}} [1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[3,1,4,5,8,2,6,7]=>{{1,2,3,4,5,6,7,8}} [1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[3,1,4,5,7,2,8,6]=>{{1,2,3,4,5,6,7,8}} [1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[3,1,4,5,6,8,2,7]=>{{1,2,3,4,5,6,7,8}} [1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[3,1,4,5,6,7,8,2]=>{{1,2,3,4,5,6,7,8}} [2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,8,1,3,4,5,6,7]=>{{1,2,3,4,5,6,7,8}} [2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[2,7,1,3,4,5,8,6]=>{{1,2,3,4,5,6,7,8}} [2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[2,6,1,3,4,8,5,7]=>{{1,2,3,4,5,6,7,8}} [2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[2,6,1,3,4,7,8,5]=>{{1,2,3,4,5,6,7,8}} [2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[2,5,1,3,8,4,6,7]=>{{1,2,3,4,5,6,7,8}} [2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[2,5,1,3,7,4,8,6]=>{{1,2,3,4,5,6,7,8}} [2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[2,5,1,3,6,8,4,7]=>{{1,2,3,4,5,6,7,8}} [2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[2,5,1,3,6,7,8,4]=>{{1,2,3,4,5,6,7,8}} [2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,4,1,8,3,5,6,7]=>{{1,2,3,4,5,6,7,8}} [2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[2,4,1,7,3,5,8,6]=>{{1,2,3,4,5,6,7,8}} [2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,4,1,6,3,8,5,7]=>{{1,2,3,4,5,6,7,8}} [2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,4,1,6,3,7,8,5]=>{{1,2,3,4,5,6,7,8}} [2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,4,1,5,8,3,6,7]=>{{1,2,3,4,5,6,7,8}} [2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[2,4,1,5,7,3,8,6]=>{{1,2,3,4,5,6,7,8}} [2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[2,4,1,5,6,8,3,7]=>{{1,2,3,4,5,6,7,8}} [2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,4,1,5,6,7,8,3]=>{{1,2,3,4,5,6,7,8}} [3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[2,3,8,1,4,5,6,7]=>{{1,2,3,4,5,6,7,8}} [3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[2,3,7,1,4,5,8,6]=>{{1,2,3,4,5,6,7,8}} [3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[2,3,6,1,4,8,5,7]=>{{1,2,3,4,5,6,7,8}} [3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[2,3,6,1,4,7,8,5]=>{{1,2,3,4,5,6,7,8}} [3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[2,3,5,1,8,4,6,7]=>{{1,2,3,4,5,6,7,8}} [3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[2,3,5,1,7,4,8,6]=>{{1,2,3,4,5,6,7,8}} [3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[2,3,5,1,6,8,4,7]=>{{1,2,3,4,5,6,7,8}} [3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[2,3,5,1,6,7,8,4]=>{{1,2,3,4,5,6,7,8}} [4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[2,3,4,8,1,5,6,7]=>{{1,2,3,4,5,6,7,8}} [4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[2,3,4,7,1,5,8,6]=>{{1,2,3,4,5,6,7,8}} [4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[2,3,4,6,1,8,5,7]=>{{1,2,3,4,5,6,7,8}} [4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[2,3,4,6,1,7,8,5]=>{{1,2,3,4,5,6,7,8}} [5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[2,3,4,5,8,1,6,7]=>{{1,2,3,4,5,6,7,8}} [5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[2,3,4,5,7,1,8,6]=>{{1,2,3,4,5,6,7,8}} [6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[2,3,4,5,6,8,1,7]=>{{1,2,3,4,5,6,7,8}} [7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,1]=>{{1,2,3,4,5,6,7,8}} [1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[9,1,2,3,4,5,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[8,1,2,3,4,5,6,9,7]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[7,1,2,3,4,5,9,6,8]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[7,1,2,3,4,5,8,9,6]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[6,1,2,3,4,9,5,7,8]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[6,1,2,3,4,8,5,9,7]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[6,1,2,3,4,7,9,5,8]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[6,1,2,3,4,7,8,9,5]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[5,1,2,3,9,4,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[5,1,2,3,8,4,6,9,7]=>{{1,2,3,4,5,6,7,8,9}} [1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[5,1,2,3,6,7,8,9,4]=>{{1,2,3,4,5,6,7,8,9}} [1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[4,1,2,8,3,5,6,9,7]=>{{1,2,3,4,5,6,7,8,9}} [1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[4,1,2,5,9,3,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[4,1,2,5,6,7,9,3,8]=>{{1,2,3,4,5,6,7,8,9}} [1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[4,1,2,5,6,7,8,9,3]=>{{1,2,3,4,5,6,7,8,9}} [1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[3,1,9,2,4,5,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[3,1,8,2,4,5,6,9,7]=>{{1,2,3,4,5,6,7,8,9}} [1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[3,1,5,2,7,4,9,6,8]=>{{1,2,3,4,5,6,7,8,9}} [1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[3,1,4,9,2,5,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[3,1,4,5,6,9,2,7,8]=>{{1,2,3,4,5,6,7,8,9}} [1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[3,1,4,5,6,8,2,9,7]=>{{1,2,3,4,5,6,7,8,9}} [1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[3,1,4,5,6,7,9,2,8]=>{{1,2,3,4,5,6,7,8,9}} [1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[3,1,4,5,6,7,8,9,2]=>{{1,2,3,4,5,6,7,8,9}} [2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,9,1,3,4,5,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,8,1,3,4,5,6,9,7]=>{{1,2,3,4,5,6,7,8,9}} [2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,7,1,3,4,5,9,6,8]=>{{1,2,3,4,5,6,7,8,9}} [2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[2,7,1,3,4,5,8,9,6]=>{{1,2,3,4,5,6,7,8,9}} [2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[2,6,1,3,4,9,5,7,8]=>{{1,2,3,4,5,6,7,8,9}} [2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,5,1,3,6,7,8,9,4]=>{{1,2,3,4,5,6,7,8,9}} [2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,4,1,6,3,8,5,9,7]=>{{1,2,3,4,5,6,7,8,9}} [2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[2,4,1,5,9,3,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[2,4,1,5,6,7,9,3,8]=>{{1,2,3,4,5,6,7,8,9}} [2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,4,1,5,6,7,8,9,3]=>{{1,2,3,4,5,6,7,8,9}} [3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,9,1,4,5,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[2,3,8,1,4,5,6,9,7]=>{{1,2,3,4,5,6,7,8,9}} [3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[2,3,5,1,6,7,9,4,8]=>{{1,2,3,4,5,6,7,8,9}} [3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,5,1,6,7,8,9,4]=>{{1,2,3,4,5,6,7,8,9}} [4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[2,3,4,9,1,5,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[2,3,4,6,1,7,8,9,5]=>{{1,2,3,4,5,6,7,8,9}} [5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[2,3,4,5,9,1,6,7,8]=>{{1,2,3,4,5,6,7,8,9}} [5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>[2,3,4,5,8,1,6,9,7]=>{{1,2,3,4,5,6,7,8,9}} [5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[2,3,4,5,7,1,8,9,6]=>{{1,2,3,4,5,6,7,8,9}} [6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[2,3,4,5,6,9,1,7,8]=>{{1,2,3,4,5,6,7,8,9}} [6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[2,3,4,5,6,8,1,9,7]=>{{1,2,3,4,5,6,7,8,9}} [7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[2,3,4,5,6,7,9,1,8]=>{{1,2,3,4,5,6,7,8,9}} [8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,9,1]=>{{1,2,3,4,5,6,7,8,9}} [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]=>[10,1,2,3,4,5,6,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[9,1,2,3,4,5,6,7,10,8]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[8,1,2,3,4,5,6,9,10,7]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[7,1,2,3,4,5,9,6,10,8]=>{{1,2,3,4,5,6,7,8,9,10}} [1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[7,1,2,3,4,5,8,9,10,6]=>{{1,2,3,4,5,6,7,8,9,10}} [1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[6,1,2,3,4,7,8,9,10,5]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[5,1,2,3,10,4,6,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[5,1,2,3,6,7,8,9,10,4]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[4,1,2,10,3,5,6,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[4,1,2,5,6,7,8,9,10,3]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[3,1,9,2,4,5,6,7,10,8]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[3,1,5,2,10,4,6,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [1,2,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[3,1,5,2,7,4,9,6,10,8]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[3,1,4,10,2,5,6,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [1,6,2]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[3,1,4,5,6,7,9,2,10,8]=>{{1,2,3,4,5,6,7,8,9,10}} [1,7,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[3,1,4,5,6,7,8,10,2,9]=>{{1,2,3,4,5,6,7,8,9,10}} [1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[3,1,4,5,6,7,8,9,10,2]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[2,10,1,3,4,5,6,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[2,9,1,3,4,5,6,7,10,8]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[2,8,1,3,4,5,6,10,7,9]=>{{1,2,3,4,5,6,7,8,9,10}} [2,1,6]=>[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,5,1,3,6,7,8,9,10,4]=>{{1,2,3,4,5,6,7,8,9,10}} [2,2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,4,1,6,3,8,5,10,7,9]=>{{1,2,3,4,5,6,7,8,9,10}} [2,6,1]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[2,4,1,5,6,7,8,10,3,9]=>{{1,2,3,4,5,6,7,8,9,10}} [2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,4,1,5,6,7,8,9,10,3]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[2,3,10,1,4,5,6,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,5,1,6,7,8,9,10,4]=>{{1,2,3,4,5,6,7,8,9,10}} [4,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,4,10,1,5,6,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [4,5]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,4,6,1,7,8,9,10,5]=>{{1,2,3,4,5,6,7,8,9,10}} [5,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,10,1,6,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [5,4]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[2,3,4,5,7,1,8,9,10,6]=>{{1,2,3,4,5,6,7,8,9,10}} [6,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]=>[2,3,4,5,6,10,1,7,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [6,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]=>[2,3,4,5,6,8,1,9,10,7]=>{{1,2,3,4,5,6,7,8,9,10}} [7,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>[2,3,4,5,6,7,10,1,8,9]=>{{1,2,3,4,5,6,7,8,9,10}} [7,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>[2,3,4,5,6,7,9,1,10,8]=>{{1,2,3,4,5,6,7,8,9,10}} [8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[2,3,4,5,6,7,8,10,1,9]=>{{1,2,3,4,5,6,7,8,9,10}} [9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,9,10,1]=>{{1,2,3,4,5,6,7,8,9,10}} [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]=>[11,1,2,3,4,5,6,7,8,9,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [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]=>[10,1,2,3,4,5,6,7,8,11,9]=>{{1,2,3,4,5,6,7,8,9,10,11}} [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]=>[7,1,2,3,4,5,11,6,8,9,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [1,8,1]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[3,1,4,5,6,7,8,9,11,2,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[3,1,4,5,6,7,8,9,10,11,2]=>{{1,2,3,4,5,6,7,8,9,10,11}} [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]=>[2,11,1,3,4,5,6,7,8,9,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [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]=>[2,10,1,3,4,5,6,7,8,11,9]=>{{1,2,3,4,5,6,7,8,9,10,11}} [2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,4,1,5,6,7,8,9,10,11,3]=>{{1,2,3,4,5,6,7,8,9,10,11}} [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]=>[2,3,11,1,4,5,6,7,8,9,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [3,7]=>[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,3,5,1,6,7,8,9,10,11,4]=>{{1,2,3,4,5,6,7,8,9,10,11}} [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]=>[2,3,4,11,1,5,6,7,8,9,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [4,6]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,4,6,1,7,8,9,10,11,5]=>{{1,2,3,4,5,6,7,8,9,10,11}} [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]=>[2,3,4,5,11,1,6,7,8,9,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [5,5]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,4,5,7,1,8,9,10,11,6]=>{{1,2,3,4,5,6,7,8,9,10,11}} [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]=>[2,3,4,5,6,11,1,7,8,9,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [6,4]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[2,3,4,5,6,8,1,9,10,11,7]=>{{1,2,3,4,5,6,7,8,9,10,11}} [7,1,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]=>[2,3,4,5,6,7,11,1,8,9,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [7,3]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]=>[2,3,4,5,6,7,9,1,10,11,8]=>{{1,2,3,4,5,6,7,8,9,10,11}} [8,1,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]=>[2,3,4,5,6,7,8,11,1,9,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [8,2]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>[2,3,4,5,6,7,8,10,1,11,9]=>{{1,2,3,4,5,6,7,8,9,10,11}} [9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[2,3,4,5,6,7,8,9,11,1,10]=>{{1,2,3,4,5,6,7,8,9,10,11}} [10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,9,10,11,1]=>{{1,2,3,4,5,6,7,8,9,10,11}} [1,10]=>[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[3,1,4,5,6,7,8,9,10,11,12,2]=>{{1,2,3,4,5,6,7,8,9,10,11,12}} [10,1]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]=>[2,3,4,5,6,7,8,9,10,12,1,11]=>{{1,2,3,4,5,6,7,8,9,10,11,12}} [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]=>[11,1,2,3,4,5,6,7,8,9,12,10]=>{{1,2,3,4,5,6,7,8,9,10,11,12}} [1,2,2,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[3,1,5,2,7,4,9,6,11,8,12,10]=>{{1,2,3,4,5,6,7,8,9,10,11,12}}
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].