nonrecursive set of parentheses related to number theory

[José Manuel Rodríguez Caballero]

Dear FOM members,
  In his famous book on elliptic functions [1], Camille Jordan mentioned
how to find the expansion of the inverse of a theta function, which today
we know it is the generating function of some topological invariants of the
Hilbert scheme of points on the two-dimensional torus [2]. This function,
expressed as (4.1.4) in [2] and as (1) in [3], can be expressed as a power
series on a variable, where the coefficients are Laurent polynomials on the
other variable. Here is the code in Mathematica for a partial expansion of
this function

Simplify[Series[Product[(1-t^n)^2/((1-q*t^n)*(1-(1/q)*t^n)),{n,
1,20}],{t,0,10}]/(1-q)]

The interesting point is that the coefficients of the Laurent polynomials
are +1 and -1. Substituting +1 by the opening parenthesis and minus one by
the closing parenthesis, we obtain a well-balanced parenthesis (Dyck word)
associated with any natural number. For example, the coefficient of t^6 is

1/q^6 + 1/q  - 1 - q^5

which corresponds to the Dyck word

DyckWord[6]  = ( ( ) ).

Here is a code in order to generate the Dyck word associated with n without
computing the series:

DyckWord[n_]:=StringJoin[If[Mod[#,2]==1,"(",")"]&/@Select[Divisors[#],Function[d,(Mod[d,2]==1)
|| (Mod[#/d,2]==1)] ]&[2*n]]

Given a Dyck word, it is not clear whether or not there is at least one
natural number corresponding to it. For example, the first natural numbers
corresponding to the Dyck word ( ( ( ) ( ) ( ) ) ) are

Select[Range[1000000], DyckWord[#]=="((()()()))"&]
= {648,10000,20000,76832,153664,307328,937024}

but as far as I know, there is no natural number corresponding to the Dyck
work ( ( ) ) ( ) ( ( ) ), since

Select[Range[1000000], DyckWord[#]=="(())()(())"&]  = { }

It may be the case that the set of Dyck words corresponding to at least a
natural number is nonrecursive. If this is the case, then several
topological properties of the Hilbert scheme of points on the torus may be
undecidable, since they are connected to Jordan's function mentioned above.

Kind regards,
Jose M.

References:
[1]  C. Jordan, Fonctions elliptiques: Calcul Integral, Springer-Verlag,
Paris, 1894.

[2] Hausel, Tamás, Emmanuel Letellier, and Fernando Rodriguez-Villegas.
"Arithmetic harmonic analysis on character and quiver varieties II."
Advances in Mathematics 234 (2013): 85-128.
URL = https://www.sciencedirect.com/science/article/pii/S0001870812004008

[3] Caballero, Jose Manuel Rodriguez. "Jordan's Expansion of the Reciprocal
of Theta Functions and 2-densely Divisible Numbers." *Integers* 20 (2020):
A2.
URL = http://math.colgate.edu/~integers/u2/u2.pdf
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