Consider how to relate the Catalan generating function to a complex number and a context-free grammar.
Consider their being k break points in the terms of the kth Mandelbrot polynomial, consider them as flip points for going between outside and inside.
How does binary tree height correspond to the difference between left-right parentheses?
Look at q-catalan numbers. How do they relate to parentheses discrepancy?
What are the q-t-catalan numbers, what do they count, and how do they relate to the Macdonald polynomials?
Consider how to relate the boundedness of the generating function to the idea that there is a finite mismatch between left and right parentheses, whereas if the number of left parentheses grows without bound, then the function diverges to infinity.
Consider what we know about the boundedness of the generating function in the complex plane.
Dievas: walks on binary trees (C,I,U). Ar tai tas pats kaip "ordered binary trees"? Ar juos skaičiuoja Catalan skaičiai?
Jeigu pridėti c + d (delta) ką sužinotumėme apie mažą paklaidą?
Consider the Mandelbrot polynomials from either end. The lower coefficients yield the Catalan polynomials, and the higher coefficients yield?
The issue of memory required by automata - that this is "explicit" truth needed to supplement "implicit" truth.
Analytic Combinatorics by Philippe Flajolet, Robert Sedgewick, 2009. They briefly mention the connection between Mandelbrot sets and Catalan numbers as noting "the speed of convergence" for "the generating functions of binary trees of bounded height". Look for Catalan numbers:
|z| ≤ r < 1/4,
simple majorant series considerations show that the convergence
is uniformly geometric. When
z ≥ s > 1/4
it can be checked that the yh(z) grow doubly exponentially. What happens in-between, in a delta–domain, needs to be quantified. We do so following Flajolet, Gao, Odlyzko, and Richmond [230, 246]. (They analyze the convergence in the interesting domain.)
It seems that the Catalan numbers serve to count both of these cases:
Whereas "all binary trees" (all parentheses) are given by binomial theorem, as can be seen by expanding ( + ) to the N.
Mandelbrot set rule: The simplest "mixing rule": "Add the input (complex vector c), add the output ("multiply by 2" by complex rotating and squaring)."
The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics (University Lecture Series) by James Haglund. With an appendix on the MacDonald polynomials (linked to root weights).
Square root as a sign of "dual" mid-point (reflection point) in a duality.