The first few most recent pending maps:

Mp00000: Permutations catalanizationPermutations
The catalanization of a permutation.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.
(Please do not accept yet, I just record it here and wait for the paper to be published.)
Sage code
def mapping(pi): 
    Pi = Permutations(len(pi)) 
    return prod([Pi(Permutation(t)) for t in sorted(pi.inversions())], 
Feb 19, 2021 at 14:02 by Christian Stump
Feb 19, 2021 at 14:02 by Christian Stump
Sage cell
Please click on evaluate and search for map before submitting your map!