There are 28 maps with codomain Permutations in the database. There are possibly some more waiting for verification.

Mp00002Alternating sign matrices ⟶ Permutations
to left key permutation
   Binary trees (2 maps)
Mp00014Binary trees ⟶ Permutations
to 132-avoiding permutation
Mp00017Binary trees ⟶ Permutations
to 312-avoiding permutation
   Dyck paths (6 maps)
Mp00023Dyck paths ⟶ Permutations
to non-crossing permutation
Mp00024Dyck paths ⟶ Permutations
to 321-avoiding permutation
Mp00025Dyck paths ⟶ Permutations
to 132-avoiding permutation
Mp00031Dyck paths ⟶ Permutations
to 312-avoiding permutation
Mp00119Dyck paths ⟶ Permutations
to 321-avoiding permutation (Krattenthaler)
Mp00129Dyck paths ⟶ Permutations
to 321-avoiding permutation (Billey-Jockusch-Stanley)
   Parking functions (2 maps)
Mp00053Parking functions ⟶ Permutations
to car permutation
Mp00055Parking functions ⟶ Permutations
to labelling permutation
   Perfect matchings (1 maps)
Mp00058Perfect matchings ⟶ Permutations
to permutation
   Permutations (13 maps)
Mp00062Permutations ⟶ Permutations
Lehmer-code to major-code bijection
Mp00064Permutations ⟶ Permutations
Mp00066Permutations ⟶ Permutations
Mp00067Permutations ⟶ Permutations
Foata bijection
Mp00068Permutations ⟶ Permutations
Simion-Schmidt map
Mp00069Permutations ⟶ Permutations
Mp00073Permutations ⟶ Permutations
major-index to inversion-number bijection
Mp00086Permutations ⟶ Permutations
first fundamental transformation
Mp00087Permutations ⟶ Permutations
inverse first fundamental transformation
Mp00088Permutations ⟶ Permutations
Kreweras complement
Mp00089Permutations ⟶ Permutations
Inverse Kreweras complement
Mp00090Permutations ⟶ Permutations
cycle-as-one-line notation
Mp00126Permutations ⟶ Permutations
cactus evacuation
   Semistandard tableaux (1 maps)
Mp00075Semistandard tableaux ⟶ Permutations
reading word permutation
   Set partitions (1 maps)
Mp00080Set partitions ⟶ Permutations
to permutation
   Standard tableaux (1 maps)
Mp00081Standard tableaux ⟶ Permutations
reading word permutation