Does it work for anyone? If no, would you like to see an explicit choice for the bijection in Mma? Here is one, checked up to n=10: CBCW[n_] := Permutations[Join[Table[L, {n}], Table[R, {n}]]] CATW[n_] := Select[CBCW[n], Apply[And, Map[# >= 0 &, Re[ FoldList[Plus, 0, # /. {R -> I + 1, L -> I - 1}]]]] &] CATW2[n_] := CATW[n] /. {L -> R, R -> L} LRW[n_] := Complement[CBCW[n], Join[CATW[n], CATW2[n]]] DIFF2[n_] := Cases[LRW[n], {__, L}] DIFF[n_] := If[n < 2, {}, Append[#, L] & /@ Permutations[ Join[Table[L, {n - 2}], Table[R, {n + 1}]]]] Biject[WalkWord_] := With[{break = Position[Re[FoldList[Plus, -2, WalkWord /. {R -> I + 1, L -> I - 1}]], -1][[1, 1]]}, Join[WalkWord[[1 ;; break - 1]] /. {L -> R, R -> L}, WalkWord[[break ;; -1]]]] Apply[And, Length[Intersection[DIFF2[#], Map[Biject, DIFF[#]]] ] == Length[DIFF2[#]] & /@ Range[0, 10]] Out[]:= True And I already explained why it should work, because the set-counting equations reduce to the def. for Pascal's triangle. My personal opinion: This tactic is definitely better than "standards mimicry", but it is it original? I don't know. It's easy to doubt, with so many thousands of works on Catalan, but again, this bijection is new to me. (Admittedly, I did not have time to check R.P. Stanley's entire oeuvre.) So what is it, speechless in Appreciation or in Prejudice? Can we get Sloane or Arndt, or another knowledgeable combinatorialist to give an opinion? (No, I do not count F. Lunnon's silly comment from yesterday) It should not be too much to ask for, after I gave a nice review of Mike Stay's preferred AI article yesterday. --Brad On Sat, Sep 28, 2019 at 6:56 PM Brad Klee <bradklee@gmail.com> wrote:
Here is an excerpt from one of my unpublished manuscripts:
https://github.com/bradklee/Docs/blob/master/CatalanExcerpt.pdf
It describes a derivation of the Catalan numbers, which I have not seen elsewhere. Does this proof work for everyone? Has anyone seen it written down elsewhere? How do you think it compares to using André's reflection method?
Thanks --Brad