Hi,

I am currently making some revisions to dyck_word.py and collecting some 
changes that were made in patches by Christian and Florent.  I just pushed 
some changes on the combinat server in case there are users who are 
interested and would like to add feedback about how certain functions are 
handled.

I'm writing to revive a topic that came up before in the following thread:
https://groups.google.com/d/msg/sage-combinat-devel/wpRtfFGOaq4/Vs8Jc0qco60J

Now we have something concrete to discuss because there are many injections 
from Dyck words to other combinatorial objects and they are implemented as:
D.to_132_avoiding_permutation
D.to_312_avoiding_permutation
D.to_321_avoiding_permutation
D.to_Catalan_code
D.to_area_sequence
D.to_non_decreasing_parking_function
D.to_noncrossing_partition
D.to_noncrossing_permutation
D.to_ordered_tree
D.to_pair_of_standard_tableaux
D.to_partition
D.to_path_string
D.to_standard_tableau
D.to_triangulation

The thing is that for some of these methods there are several 
injections/bijections.  For example, there may be many maps from Dyck words 
to xyz-avoiding permutations or noncrossing_partitions.  Do you have 
suggestions about how to handle the duplicates?

Example: I want to add another to_noncrossing_partition method, but I 
wouldn't want to call it
D.to_noncrossing_partition_1
vs.
D.to_noncrossing_partition_2
since neither name is very helpful.

Christian had one way of handling this that I thought was OK, but I thought 
not ideal.
D.to_permutation("Bandlow-Kilpatrick")
which is now
D.to_312_avoiding_permutation()

and

D.to_permutation("Krattenthaler")
which is now
D.to_132_avoiding_permutation()

I thought it was best to break up to_permutation because these maps weren't 
the same set of permutations, but there are different bijections for the 
exact same sets that are different and perhaps both are interesting because 
they preserve different statistics.  I would like to still these variations 
in maps, but I would like suggestions about how to resolve the name space 
if possible.

Christian may have more to add, because he has thought about this more than 
I have and he is also motivated by the combinatorial statistics patch.

-Mike

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