Identifier
Mp00231: Integer compositions bounce pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00151: Permutations to cycle type Set partitions
Images
=>
Cc0005;cc-rep-1Cc0009;cc-rep-3
[1]=>[1,0]=>[1]=>{{1}} [1,1]=>[1,0,1,0]=>[1,2]=>{{1},{2}} [2]=>[1,1,0,0]=>[2,1]=>{{1,2}} [1,1,1]=>[1,0,1,0,1,0]=>[1,2,3]=>{{1},{2},{3}} [1,2]=>[1,0,1,1,0,0]=>[1,3,2]=>{{1},{2,3}} [2,1]=>[1,1,0,0,1,0]=>[2,1,3]=>{{1,2},{3}} [3]=>[1,1,1,0,0,0]=>[3,1,2]=>{{1,2,3}} [1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[1,2,3,4]=>{{1},{2},{3},{4}} [1,1,2]=>[1,0,1,0,1,1,0,0]=>[1,2,4,3]=>{{1},{2},{3,4}} [1,2,1]=>[1,0,1,1,0,0,1,0]=>[1,3,2,4]=>{{1},{2,3},{4}} [1,3]=>[1,0,1,1,1,0,0,0]=>[1,4,2,3]=>{{1},{2,3,4}} [2,1,1]=>[1,1,0,0,1,0,1,0]=>[2,1,3,4]=>{{1,2},{3},{4}} [2,2]=>[1,1,0,0,1,1,0,0]=>[2,1,4,3]=>{{1,2},{3,4}} [3,1]=>[1,1,1,0,0,0,1,0]=>[3,1,2,4]=>{{1,2,3},{4}} [4]=>[1,1,1,1,0,0,0,0]=>[4,1,2,3]=>{{1,2,3,4}} [1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5]=>{{1},{2},{3},{4},{5}} [1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[1,2,3,5,4]=>{{1},{2},{3},{4,5}} [1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[1,2,4,3,5]=>{{1},{2},{3,4},{5}} [1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[1,2,5,3,4]=>{{1},{2},{3,4,5}} [1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[1,3,2,4,5]=>{{1},{2,3},{4},{5}} [1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4]=>{{1},{2,3},{4,5}} [1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[1,4,2,3,5]=>{{1},{2,3,4},{5}} [1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[1,5,2,3,4]=>{{1},{2,3,4,5}} [2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[2,1,3,4,5]=>{{1,2},{3},{4},{5}} [2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[2,1,3,5,4]=>{{1,2},{3},{4,5}} [2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,5]=>{{1,2},{3,4},{5}} [2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[2,1,5,3,4]=>{{1,2},{3,4,5}} [3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[3,1,2,4,5]=>{{1,2,3},{4},{5}} [3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[3,1,2,5,4]=>{{1,2,3},{4,5}} [4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[4,1,2,3,5]=>{{1,2,3,4},{5}} [5]=>[1,1,1,1,1,0,0,0,0,0]=>[5,1,2,3,4]=>{{1,2,3,4,5}} [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},{2},{3},{4},{5},{6}} [1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,6,5]=>{{1},{2},{3},{4},{5,6}} [1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,5,4,6]=>{{1},{2},{3},{4,5},{6}} [1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,6,4,5]=>{{1},{2},{3},{4,5,6}} [1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,5,6]=>{{1},{2},{3,4},{5},{6}} [1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5]=>{{1},{2},{3,4},{5,6}} [1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,5,3,4,6]=>{{1},{2},{3,4,5},{6}} [1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,6,3,4,5]=>{{1},{2},{3,4,5,6}} [1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,4,5,6]=>{{1},{2,3},{4},{5},{6}} [1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[1,3,2,4,6,5]=>{{1},{2,3},{4},{5,6}} [1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,6]=>{{1},{2,3},{4,5},{6}} [1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[1,3,2,6,4,5]=>{{1},{2,3},{4,5,6}} [1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[1,4,2,3,5,6]=>{{1},{2,3,4},{5},{6}} [1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[1,4,2,3,6,5]=>{{1},{2,3,4},{5,6}} [1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[1,5,2,3,4,6]=>{{1},{2,3,4,5},{6}} [1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,6,2,3,4,5]=>{{1},{2,3,4,5,6}} [2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6]=>{{1,2},{3},{4},{5},{6}} [2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,6,5]=>{{1,2},{3},{4},{5,6}} [2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,3,5,4,6]=>{{1,2},{3},{4,5},{6}} [2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,1,3,6,4,5]=>{{1,2},{3},{4,5,6}} [2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,1,4,3,5,6]=>{{1,2},{3,4},{5},{6}} [2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5]=>{{1,2},{3,4},{5,6}} [2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,1,5,3,4,6]=>{{1,2},{3,4,5},{6}} [2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,6,3,4,5]=>{{1,2},{3,4,5,6}} [3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[3,1,2,4,5,6]=>{{1,2,3},{4},{5},{6}} [3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[3,1,2,4,6,5]=>{{1,2,3},{4},{5,6}} [3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[3,1,2,5,4,6]=>{{1,2,3},{4,5},{6}} [3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[3,1,2,6,4,5]=>{{1,2,3},{4,5,6}} [4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[4,1,2,3,5,6]=>{{1,2,3,4},{5},{6}} [4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[4,1,2,3,6,5]=>{{1,2,3,4},{5,6}} [5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[5,1,2,3,4,6]=>{{1,2,3,4,5},{6}} [6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[6,1,2,3,4,5]=>{{1,2,3,4,5,6}} [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},{2},{3},{4},{5},{6},{7}} [1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,7,6]=>{{1},{2},{3},{4},{5},{6,7}} [1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,4,6,5,7]=>{{1},{2},{3},{4},{5,6},{7}} [1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,7,5,6]=>{{1},{2},{3},{4},{5,6,7}} [1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,3,5,4,6,7]=>{{1},{2},{3},{4,5},{6},{7}} [1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,3,5,4,7,6]=>{{1},{2},{3},{4,5},{6,7}} [1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,3,6,4,5,7]=>{{1},{2},{3},{4,5,6},{7}} [1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,7,4,5,6]=>{{1},{2},{3},{4,5,6,7}} [1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,4,3,5,6,7]=>{{1},{2},{3,4},{5},{6},{7}} [1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,2,4,3,5,7,6]=>{{1},{2},{3,4},{5},{6,7}} [1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5,7]=>{{1},{2},{3,4},{5,6},{7}} [1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,4,3,7,5,6]=>{{1},{2},{3,4},{5,6,7}} [1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,2,5,3,4,6,7]=>{{1},{2},{3,4,5},{6},{7}} [1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,2,5,3,4,7,6]=>{{1},{2},{3,4,5},{6,7}} [1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,6,3,4,5,7]=>{{1},{2},{3,4,5,6},{7}} [1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,7,3,4,5,6]=>{{1},{2},{3,4,5,6,7}} [1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,3,2,4,5,6,7]=>{{1},{2,3},{4},{5},{6},{7}} [1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,3,2,4,5,7,6]=>{{1},{2,3},{4},{5},{6,7}} [1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,6,5,7]=>{{1},{2,3},{4},{5,6},{7}} [1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,3,2,4,7,5,6]=>{{1},{2,3},{4},{5,6,7}} [1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,4,6,7]=>{{1},{2,3},{4,5},{6},{7}} [1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,7,6]=>{{1},{2,3},{4,5},{6,7}} [1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,3,2,6,4,5,7]=>{{1},{2,3},{4,5,6},{7}} [1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,7,4,5,6]=>{{1},{2,3},{4,5,6,7}} [1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,4,2,3,5,6,7]=>{{1},{2,3,4},{5},{6},{7}} [1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,4,2,3,5,7,6]=>{{1},{2,3,4},{5},{6,7}} [1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,4,2,3,6,5,7]=>{{1},{2,3,4},{5,6},{7}} [1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,4,2,3,7,5,6]=>{{1},{2,3,4},{5,6,7}} [1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,5,2,3,4,6,7]=>{{1},{2,3,4,5},{6},{7}} [1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,5,2,3,4,7,6]=>{{1},{2,3,4,5},{6,7}} [1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,6,2,3,4,5,7]=>{{1},{2,3,4,5,6},{7}} [1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,7,2,3,4,5,6]=>{{1},{2,3,4,5,6,7}} [2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6,7]=>{{1,2},{3},{4},{5},{6},{7}} [2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,7,6]=>{{1,2},{3},{4},{5},{6,7}} [2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,3,4,6,5,7]=>{{1,2},{3},{4},{5,6},{7}} [2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[2,1,3,4,7,5,6]=>{{1,2},{3},{4},{5,6,7}} [2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[2,1,3,5,4,6,7]=>{{1,2},{3},{4,5},{6},{7}} [2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[2,1,3,5,4,7,6]=>{{1,2},{3},{4,5},{6,7}} [2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[2,1,3,6,4,5,7]=>{{1,2},{3},{4,5,6},{7}} [2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,3,7,4,5,6]=>{{1,2},{3},{4,5,6,7}} [2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,3,5,6,7]=>{{1,2},{3,4},{5},{6},{7}} [2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[2,1,4,3,5,7,6]=>{{1,2},{3,4},{5},{6,7}} [2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,7]=>{{1,2},{3,4},{5,6},{7}} [2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,7,5,6]=>{{1,2},{3,4},{5,6,7}} [2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,5,3,4,6,7]=>{{1,2},{3,4,5},{6},{7}} [2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,5,3,4,7,6]=>{{1,2},{3,4,5},{6,7}} [2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,6,3,4,5,7]=>{{1,2},{3,4,5,6},{7}} [2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,7,3,4,5,6]=>{{1,2},{3,4,5,6,7}} [3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[3,1,2,4,5,6,7]=>{{1,2,3},{4},{5},{6},{7}} [3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[3,1,2,4,5,7,6]=>{{1,2,3},{4},{5},{6,7}} [3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[3,1,2,4,6,5,7]=>{{1,2,3},{4},{5,6},{7}} [3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[3,1,2,4,7,5,6]=>{{1,2,3},{4},{5,6,7}} [3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[3,1,2,5,4,6,7]=>{{1,2,3},{4,5},{6},{7}} [3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[3,1,2,5,4,7,6]=>{{1,2,3},{4,5},{6,7}} [3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[3,1,2,6,4,5,7]=>{{1,2,3},{4,5,6},{7}} [3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[3,1,2,7,4,5,6]=>{{1,2,3},{4,5,6,7}} [4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[4,1,2,3,5,6,7]=>{{1,2,3,4},{5},{6},{7}} [4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[4,1,2,3,5,7,6]=>{{1,2,3,4},{5},{6,7}} [4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[4,1,2,3,6,5,7]=>{{1,2,3,4},{5,6},{7}} [4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[4,1,2,3,7,5,6]=>{{1,2,3,4},{5,6,7}} [5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[5,1,2,3,4,6,7]=>{{1,2,3,4,5},{6},{7}} [5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[5,1,2,3,4,7,6]=>{{1,2,3,4,5},{6,7}} [6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[6,1,2,3,4,5,7]=>{{1,2,3,4,5,6},{7}} [7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[7,1,2,3,4,5,6]=>{{1,2,3,4,5,6,7}} [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},{2},{3},{4},{5},{6},{7},{8}} [1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,6,5,7,8]=>{{1},{2},{3,4},{5,6},{7},{8}} [1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,5,7,6,8]=>{{1},{2,3},{4},{5},{6,7},{8}} [1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[1,4,2,3,7,5,6,8]=>{{1},{2,3,4},{5,6,7},{8}} [2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,6,8,7]=>{{1,2},{3},{4},{5},{6},{7,8}} [2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5,8,7]=>{{1,2},{3,4},{5,6},{7,8}} [3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[3,1,2,4,5,8,6,7]=>{{1,2,3},{4},{5},{6,7,8}} [4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[4,1,2,3,8,5,6,7]=>{{1,2,3,4},{5,6,7,8}}
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
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].