- FindStat

Identifier

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions

Images

[(1,2)] => [2,1] => [2,1] => [1,1,0,0] => [2]

[(1,2),(3,4)] => [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0] => [4]

[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => [4]

[(1,4),(2,3)] => [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => [4]

[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0] => [6]

[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0] => [6]

[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [3,4,2,6,1,5] => [1,1,1,0,1,0,0,1,1,0,0,0] => [6]

[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [5,3,6,2,1,4] => [1,1,1,1,1,0,0,1,0,0,0,0] => [6]

[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [5,3,6,2,4,1] => [1,1,1,1,1,0,0,1,0,0,0,0] => [6]

[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [4,5,2,6,3,1] => [1,1,1,1,0,1,0,0,1,0,0,0] => [6]

[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [4,5,2,1,6,3] => [1,1,1,1,0,1,0,0,0,1,0,0] => [6]

[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [1,1,1,1,0,0,1,0,0,1,0,0] => [6]

[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [5,3,1,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0] => [6]

[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6]

[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [5,6,2,4,1,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6]

[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [3,5,6,1,4,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [6]

[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [4,1,5,6,3,2] => [1,1,1,1,0,0,1,0,1,0,0,0] => [6]

[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [4,5,1,6,3,2] => [1,1,1,1,0,1,0,0,1,0,0,0] => [6]

[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => [6]

[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0] => [8]

[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [3,1,4,6,8,2,5,7] => [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0] => [8]

[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [3,4,2,6,8,1,5,7] => [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0] => [8]

[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [5,3,6,2,8,1,4,7] => [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0] => [8]

[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [5,3,6,2,4,8,1,7] => [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0] => [8]

[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [5,7,3,8,2,4,1,6] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [8]

[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [5,7,3,8,2,4,6,1] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [8]

[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [7,4,2,5,8,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [8]

[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [7,4,2,5,8,3,1,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [8]

[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [4,5,2,6,3,8,1,7] => [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0] => [8]

[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [4,5,2,1,6,8,3,7] => [1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0] => [8]

[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [4,1,5,2,6,8,3,7] => [1,1,1,1,0,0,1,0,0,1,0,1,1,0,0,0] => [8]

[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [5,3,1,6,8,2,4,7] => [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0] => [8]

[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [5,6,2,8,1,3,4,7] => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0] => [8]

[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [5,6,2,4,8,1,3,7] => [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0] => [8]

[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [3,5,6,1,4,8,2,7] => [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0] => [8]

[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [4,1,5,6,3,8,2,7] => [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0] => [8]

[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [4,5,1,6,3,8,2,7] => [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0] => [8]

[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [4,5,6,3,2,8,1,7] => [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0] => [8]

[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [7,4,5,3,8,2,1,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [8]

[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [7,4,5,3,8,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [8]

[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [6,7,4,8,3,2,5,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [8]

[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [6,7,4,8,3,2,1,5] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [8]

[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [6,7,4,1,8,3,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [8]

[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [7,4,5,3,1,8,2,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [8]

[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [4,7,1,5,8,3,2,6] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0] => [8]

[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [5,3,7,8,1,4,2,6] => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0] => [8]

[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [7,5,8,2,4,1,3,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [8]

[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [7,5,8,2,4,6,1,3] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [8]

[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [7,3,5,8,1,4,6,2] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [8]

[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [4,5,1,7,8,3,6,2] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0] => [8]

[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [7,4,1,5,8,3,6,2] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [8]

[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [6,7,4,3,8,1,5,2] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [8]

[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [6,7,4,8,3,1,5,2] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [8]

[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [6,7,4,8,3,5,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [8]

[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [5,6,7,3,8,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [8]

[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [5,6,7,3,1,8,4,2] => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0] => [8]

[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [5,6,1,7,3,8,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => [8]

[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [5,1,6,7,3,8,4,2] => [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0] => [8]

[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [6,4,7,1,8,3,5,2] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [8]

[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [6,7,8,3,1,4,5,2] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [8]

[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [6,7,2,8,4,5,1,3] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [8]

[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [6,7,2,8,4,1,3,5] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [8]

[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [3,6,1,7,8,4,2,5] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0] => [8]

[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [6,1,7,4,8,3,2,5] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [8]

[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [5,1,6,7,3,2,8,4] => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [8]

[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [5,6,1,7,3,2,8,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [8]

[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [5,6,7,3,2,1,8,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0] => [8]

[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [5,6,7,3,2,8,4,1] => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0] => [8]

[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [5,6,2,7,3,8,4,1] => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => [8]

[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [5,6,2,7,3,1,8,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [8]

[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [5,6,2,1,7,3,8,4] => [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0] => [8]

[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0] => [8]

[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [6,1,7,4,2,8,3,5] => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0] => [8]

[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [3,6,1,7,2,8,4,5] => [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0] => [8]

[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [6,7,2,1,8,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [8]

[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [7,5,8,2,1,3,4,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [8]

[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [5,7,3,1,8,2,4,6] => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [8]

[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [4,7,1,5,2,8,3,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0] => [8]

[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [7,4,2,5,1,8,3,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [8]

[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [6,7,4,2,1,8,3,5] => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0] => [8]

[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [6,7,4,2,8,3,1,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [8]

[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [6,7,4,2,8,3,5,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [8]

[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [3,6,7,2,8,4,5,1] => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => [8]

[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [3,6,7,2,8,4,1,5] => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => [8]

[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [3,6,7,2,1,8,4,5] => [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0] => [8]

[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [5,7,3,8,2,1,4,6] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [8]

[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [3,4,7,8,2,1,5,6] => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [8]

[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [3,4,7,1,8,2,5,6] => [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0] => [8]

[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [7,8,2,4,1,3,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [8]

[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [7,8,2,4,6,1,3,5] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [8]

[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [3,7,8,1,4,6,2,5] => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [8]

[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [3,4,7,8,2,6,1,5] => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [8]

[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [5,7,3,8,2,6,1,4] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [8]

[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [3,6,7,2,8,5,1,4] => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => [8]

[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [3,6,7,8,5,2,1,4] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0] => [8]

[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [3,6,7,8,5,2,4,1] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0] => [8]

[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [6,4,7,8,2,5,3,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [8]

[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [6,4,7,8,2,1,5,3] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [8]

[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [6,4,7,2,8,1,5,3] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [8]

[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [4,5,2,7,8,1,6,3] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0] => [8]

[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [4,1,5,7,8,2,6,3] => [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0] => [8]

>>> Load all 147 entries. <<<

[(1,3),(2,5),(4,8),(6,7)] => [3,5,1,7,2,8,6,4] => [5,3,7,8,1,6,2,4] => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0] => [8]

[(1,2),(3,5),(4,8),(6,7)] => [2,1,5,7,3,8,6,4] => [7,5,8,2,6,1,3,4] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [8]

[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,7,8,3,5,4] => [6,7,2,8,5,1,3,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [8]

[(1,3),(2,6),(4,8),(5,7)] => [3,6,1,7,8,2,5,4] => [3,6,1,7,8,5,2,4] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0] => [8]

[(1,4),(2,6),(3,8),(5,7)] => [4,6,7,1,8,2,5,3] => [6,4,1,7,8,2,5,3] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [8]

[(1,5),(2,6),(3,8),(4,7)] => [5,6,7,8,1,2,4,3] => [5,1,6,2,7,8,4,3] => [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0] => [8]

[(1,6),(2,5),(3,8),(4,7)] => [5,6,7,8,2,1,4,3] => [5,6,2,1,7,8,4,3] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0] => [8]

[(1,7),(2,5),(3,8),(4,6)] => [5,6,7,8,2,4,1,3] => [5,6,2,7,1,8,4,3] => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => [8]

[(1,8),(2,5),(3,7),(4,6)] => [5,6,7,8,2,4,3,1] => [5,6,2,7,8,4,3,1] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [8]

[(1,8),(2,6),(3,7),(4,5)] => [5,6,7,8,4,2,3,1] => [5,6,7,2,8,4,3,1] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [8]

[(1,7),(2,6),(3,8),(4,5)] => [5,6,7,8,4,2,1,3] => [5,6,7,2,1,8,4,3] => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0] => [8]

[(1,6),(2,7),(3,8),(4,5)] => [5,6,7,8,4,1,2,3] => [5,6,1,7,2,8,4,3] => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => [8]

[(1,5),(2,7),(3,8),(4,6)] => [5,6,7,8,1,4,2,3] => [5,1,6,7,2,8,4,3] => [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0] => [8]

[(1,4),(2,7),(3,8),(5,6)] => [4,6,7,1,8,5,2,3] => [6,4,7,1,8,2,5,3] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [8]

[(1,3),(2,7),(4,8),(5,6)] => [3,6,1,7,8,5,2,4] => [3,6,7,1,8,5,2,4] => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => [8]

[(1,2),(3,7),(4,8),(5,6)] => [2,1,6,7,8,5,3,4] => [6,7,8,5,2,1,3,4] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [8]

[(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => [6,7,8,5,2,4,1,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [8]

[(1,3),(2,8),(4,7),(5,6)] => [3,6,1,7,8,5,4,2] => [6,7,8,3,5,1,4,2] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [8]

[(1,4),(2,8),(3,7),(5,6)] => [4,6,7,1,8,5,3,2] => [4,6,7,8,1,5,3,2] => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0] => [8]

[(1,5),(2,8),(3,7),(4,6)] => [5,6,7,8,1,4,3,2] => [5,1,6,7,8,4,3,2] => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [8]

[(1,6),(2,8),(3,7),(4,5)] => [5,6,7,8,4,1,3,2] => [5,6,1,7,8,4,3,2] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [8]

[(1,7),(2,8),(3,6),(4,5)] => [5,6,7,8,4,3,1,2] => [5,6,7,1,8,4,3,2] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [8]

[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [8]

[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0] => [10]

[(1,2),(3,10),(4,5),(6,7),(8,9)] => [2,1,5,7,4,9,6,10,8,3] => [7,9,5,10,2,4,6,8,1,3] => [1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0] => [10]

[(1,10),(2,9),(3,4),(5,6),(7,8)] => [4,6,8,3,9,5,10,7,2,1] => [8,6,9,4,10,3,5,7,2,1] => [1,1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0,0] => [10]

[(1,10),(2,3),(4,7),(5,6),(8,9)] => [3,6,2,7,9,5,4,10,8,1] => [9,3,6,7,5,10,2,4,8,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0] => [10]

[(1,10),(2,7),(3,6),(4,5),(8,9)] => [5,6,7,9,4,3,2,10,8,1] => [9,5,6,7,4,3,10,2,8,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0] => [10]

[(1,2),(3,10),(4,7),(5,6),(8,9)] => [2,1,6,7,9,5,4,10,8,3] => [9,6,7,5,10,2,4,8,1,3] => [1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [10]

[(1,10),(2,9),(3,6),(4,5),(7,8)] => [5,6,8,9,4,3,10,7,2,1] => [8,9,5,6,4,10,3,7,2,1] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0] => [10]

[(1,2),(3,4),(5,10),(6,7),(8,9)] => [2,1,4,3,7,9,6,10,8,5] => [9,7,10,2,4,6,8,1,3,5] => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [10]

[(1,4),(2,3),(5,10),(6,7),(8,9)] => [3,4,2,1,7,9,6,10,8,5] => [9,3,4,7,10,2,6,8,1,5] => [1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [10]

[(1,2),(3,10),(4,9),(5,6),(7,8)] => [2,1,6,8,9,5,10,7,4,3] => [8,9,6,10,5,7,2,4,1,3] => [1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [10]

[(1,10),(2,9),(3,8),(4,6),(5,7)] => [6,7,8,9,10,4,5,3,2,1] => [6,7,8,9,4,10,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0] => [10]

[(1,6),(2,7),(3,8),(4,9),(5,10)] => [6,7,8,9,10,1,2,3,4,5] => [6,1,7,2,8,3,9,4,10,5] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0] => [10]

[(1,10),(2,6),(3,7),(4,8),(5,9)] => [6,7,8,9,10,2,3,4,5,1] => [6,7,2,8,3,9,4,10,5,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0] => [10]

[(1,2),(3,4),(5,6),(7,10),(8,9)] => [2,1,4,3,6,5,9,10,8,7] => [9,10,2,4,6,8,1,3,5,7] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [10]

[(1,10),(2,3),(4,5),(6,9),(7,8)] => [3,5,2,8,4,9,10,7,6,1] => [8,5,9,10,3,7,2,4,6,1] => [1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [10]

[(1,10),(2,9),(3,4),(5,8),(6,7)] => [4,7,8,3,9,10,6,5,2,1] => [7,4,8,9,10,6,3,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0] => [10]

[(1,2),(3,4),(5,10),(6,9),(7,8)] => [2,1,4,3,8,9,10,7,6,5] => [8,9,10,7,2,4,6,1,3,5] => [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [10]

[(1,4),(2,3),(5,10),(6,9),(7,8)] => [3,4,2,1,8,9,10,7,6,5] => [8,9,10,3,4,7,2,6,1,5] => [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [10]

[(1,10),(2,3),(4,9),(5,8),(6,7)] => [3,7,2,8,9,10,6,5,4,1] => [7,8,9,10,3,6,5,2,4,1] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [10]

[(1,2),(3,10),(4,9),(5,8),(6,7)] => [2,1,7,8,9,10,6,5,4,3] => [7,8,9,10,6,5,2,4,1,3] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [10]

[(1,10),(2,9),(3,8),(4,7),(5,6)] => [6,7,8,9,10,5,4,3,2,1] => [6,7,8,9,10,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [10]

[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0] => [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] => [7,1,8,2,9,3,10,4,11,5,12,6] => [1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0] => [12]

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

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00067Foata bijection.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.