Nothing equivalent on A002054? http://oeis.org/A002054 Sometimes I have trouble decoding these entries though... —Brad
On Sep 29, 2019, at 12:44 PM, Brad Klee <bradklee@gmail.com> wrote:
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
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)
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Here is an excerpt from one of my unpublished manuscripts: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