- FindStat

Identifier

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 149 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[(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}}

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

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

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

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

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

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

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

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

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

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

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

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

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

[(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}}

Map

to permutation

Description

Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.

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].