- FindStat

Identifier

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions

Images

[(1,2)] => [2,1] => {{1,2}} => {{1,2}}

[(1,2),(3,4)] => [2,1,4,3] => {{1,2},{3,4}} => {{1,2},{3,4}}

[(1,3),(2,4)] => [3,4,1,2] => {{1,3},{2,4}} => {{1,4},{2,3}}

[(1,4),(2,3)] => [3,4,2,1] => {{1,3},{2,4}} => {{1,4},{2,3}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}} => {{1,2},{3,4},{5,6},{7,8}}

[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => {{1,3},{2,4},{5,6},{7,8}} => {{1,4},{2,3},{5,6},{7,8}}

[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => {{1,3},{2,4},{5,6},{7,8}} => {{1,4},{2,3},{5,6},{7,8}}

[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => {{1,3},{2,5},{4,6},{7,8}} => {{1,6},{2,3},{4,5},{7,8}}

[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => {{1,3},{2,5},{4,6},{7,8}} => {{1,6},{2,3},{4,5},{7,8}}

[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}}

[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}}

[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => {{1,4},{2,5},{3,6},{7,8}} => {{1,6},{2,5},{3,4},{7,8}}

[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => {{1,4},{2,5},{3,6},{7,8}} => {{1,6},{2,5},{3,4},{7,8}}

[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => {{1,4},{2,5},{3,6},{7,8}} => {{1,6},{2,5},{3,4},{7,8}}

[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => {{1,3},{2,5},{4,6},{7,8}} => {{1,6},{2,3},{4,5},{7,8}}

[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => {{1,2},{3,5},{4,6},{7,8}} => {{1,2},{3,6},{4,5},{7,8}}

[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => {{1,2},{3,5},{4,6},{7,8}} => {{1,2},{3,6},{4,5},{7,8}}

[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => {{1,3},{2,5},{4,6},{7,8}} => {{1,6},{2,3},{4,5},{7,8}}

[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => {{1,4},{2,5},{3,6},{7,8}} => {{1,6},{2,5},{3,4},{7,8}}

[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => {{1,4},{2,5},{3,6},{7,8}} => {{1,6},{2,5},{3,4},{7,8}}

[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => {{1,4},{2,5},{3,6},{7,8}} => {{1,6},{2,5},{3,4},{7,8}}

[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}}

[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => {{1,2},{3,5},{4,7},{6,8}} => {{1,2},{3,8},{4,5},{6,7}}

[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => {{1,2},{3,5},{4,7},{6,8}} => {{1,2},{3,8},{4,5},{6,7}}

[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}}

[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => {{1,2},{3,6},{4,7},{5,8}} => {{1,2},{3,8},{4,7},{5,6}}

[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => {{1,2},{3,6},{4,7},{5,8}} => {{1,2},{3,8},{4,7},{5,6}}

[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => {{1,2},{3,6},{4,7},{5,8}} => {{1,2},{3,8},{4,7},{5,6}}

[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => {{1,2},{3,5},{4,7},{6,8}} => {{1,2},{3,8},{4,5},{6,7}}

[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}}

[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}}

[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => {{1,3},{2,4},{5,7},{6,8}} => {{1,4},{2,3},{5,8},{6,7}}

[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => {{1,3},{2,4},{5,7},{6,8}} => {{1,4},{2,3},{5,8},{6,7}}

[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => {{1,2},{3,4},{5,7},{6,8}} => {{1,2},{3,4},{5,8},{6,7}}

[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => {{1,2},{3,4},{5,7},{6,8}} => {{1,2},{3,4},{5,8},{6,7}}

[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => {{1,3},{2,4},{5,7},{6,8}} => {{1,4},{2,3},{5,8},{6,7}}

[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => {{1,3},{2,4},{5,7},{6,8}} => {{1,4},{2,3},{5,8},{6,7}}

[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}}

[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}}

>>> Load all 130 entries. <<<

[(1,3),(2,5),(4,8),(6,7)] => [3,5,1,7,2,8,6,4] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}}

[(1,2),(3,5),(4,8),(6,7)] => [2,1,5,7,3,8,6,4] => {{1,2},{3,5},{4,7},{6,8}} => {{1,2},{3,8},{4,5},{6,7}}

[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,7,8,3,5,4] => {{1,2},{3,6},{4,7},{5,8}} => {{1,2},{3,8},{4,7},{5,6}}

[(1,3),(2,6),(4,8),(5,7)] => [3,6,1,7,8,2,5,4] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,4),(2,6),(3,8),(5,7)] => [4,6,7,1,8,2,5,3] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,5),(2,6),(3,8),(4,7)] => [5,6,7,8,1,2,4,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,6),(2,5),(3,8),(4,7)] => [5,6,7,8,2,1,4,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,7),(2,5),(3,8),(4,6)] => [5,6,7,8,2,4,1,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,8),(2,5),(3,7),(4,6)] => [5,6,7,8,2,4,3,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,8),(2,6),(3,7),(4,5)] => [5,6,7,8,4,2,3,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,7),(2,6),(3,8),(4,5)] => [5,6,7,8,4,2,1,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,6),(2,7),(3,8),(4,5)] => [5,6,7,8,4,1,2,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,5),(2,7),(3,8),(4,6)] => [5,6,7,8,1,4,2,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,4),(2,7),(3,8),(5,6)] => [4,6,7,1,8,5,2,3] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,3),(2,7),(4,8),(5,6)] => [3,6,1,7,8,5,2,4] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,2),(3,7),(4,8),(5,6)] => [2,1,6,7,8,5,3,4] => {{1,2},{3,6},{4,7},{5,8}} => {{1,2},{3,8},{4,7},{5,6}}

[(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => {{1,2},{3,6},{4,7},{5,8}} => {{1,2},{3,8},{4,7},{5,6}}

[(1,3),(2,8),(4,7),(5,6)] => [3,6,1,7,8,5,4,2] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}}

[(1,4),(2,8),(3,7),(5,6)] => [4,6,7,1,8,5,3,2] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}}

[(1,5),(2,8),(3,7),(4,6)] => [5,6,7,8,1,4,3,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,6),(2,8),(3,7),(4,5)] => [5,6,7,8,4,1,3,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,7),(2,8),(3,6),(4,5)] => [5,6,7,8,4,3,1,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}}

[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => {{1,2},{3,4},{5,6},{7,8},{9,10}} => {{1,2},{3,4},{5,6},{7,8},{9,10}}

[(1,6),(2,7),(3,8),(4,9),(5,10)] => [6,7,8,9,10,1,2,3,4,5] => {{1,6},{2,7},{3,8},{4,9},{5,10}} => {{1,10},{2,9},{3,8},{4,7},{5,6}}

[(1,10),(2,9),(3,8),(4,7),(5,6)] => [6,7,8,9,10,5,4,3,2,1] => {{1,6},{2,7},{3,8},{4,9},{5,10}} => {{1,10},{2,9},{3,8},{4,7},{5,6}}

[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)] => [7,8,9,10,11,12,6,5,4,3,2,1] => {{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}} => {{1,12},{2,11},{3,10},{4,9},{5,8},{6,7}}

[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}} => {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}

[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)] => [7,8,9,10,11,12,1,2,3,4,5,6] => {{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}} => {{1,12},{2,11},{3,10},{4,9},{5,8},{6,7}}

Map

non-nesting-exceedence permutation

Description

The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.

Map

weak exceedance partition

Description

The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.

Map

Wachs-White-rho

Description

A transformation of set partitions due to Wachs and White.
Return the set partition of $\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Wachs and White's bijection $\rho$.
Note that our index convention differs from the convention in [1]: regarding the rook board as a lower-right triangular grid, we refer with $(i,j)$ to the cell in the $i$-th column from the right and the $j$-th row from the top.