Hi Fred, Thanks for your interest. It’s just an excerpt, so context might be difficult to pick up right away. Maximally permissive, meaning, allowing any number of arguments, in contrast to minimally permissive unary or binary functions. The term “head” is consistent usage with Wolfram language: https://reference.wolfram.com/language/ref/Head.html The theorem in question is the well know fact that Catalan numbers can be written as a multiple of central binomial coefficients: C_n = 1/(n+1)*C(2*n,n) However the claimed “proof” or at least “rigorous derivation” is also about lattice walks, and the potentially novel idea is that the set DIFFW is useful for counting the complement of CATW in CBCW. The first paragraph explains in detail how to go from sets of function compositions to sets of lattice walks. If that is too confusing for you, it is possible just to skip the first paragraph, and start reading at paragraph 2. Cheers —Brad

On Sep 28, 2019, at 8:42 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:

I couldn't understand the first line. If you must use obscure undefined notation such as "maximally permissive" and "head", please cite an introductory text where the reader might find them explained.

It remains obscure to me both what statement you intended to prove, and where your claimed proof starts and finishes.

WFL

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On Sun, Sep 29, 2019 at 12:53 AM 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Here is an excerpt from one of my unpublished manuscripts: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

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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)

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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