Act I — Motivating Example

• Max-peak-unique Dyck paths: Dyck paths with unique peak of max. height

Enumerating max-peak-unique Dyck paths

From Coefficient Extraction ...

• Ingredients:
  • $U_h(\cosh w) = \sinh((h+1) w) / \sinh w$
  • $x = \sqrt{t} / (1 + t) \iff 1/(2x) = \cosh(\frac{1}{2} \log t)$

\[ \scriptsize\Rightarrow A(x) = \sum\limits_{h\geq 0} \frac{t^h (1 - t)^2}{(1 - t^{h+1})^2} \]

... to an Inequality ...

• Bóna—DeJonge-question $\iff a_n / C_n \gt a_{n+1} / C_{n+1}$ for $n\geq 3$
• After some further manipulations, this is equivalent to...

... followed by Asymptotics!

• Split binomial sum into "central" and "tails" regions, show contribution
  of tails is negligible
• Approximate $\binom{2n}{n-k}$ in "central" region, complete tails
• Mellin for integral representation; shift line of integration + collect residues

• Not good enough: no information about error—can only say sequence is eventually decreasing...

Act II — dependent_bterms in SageMath

• $B$-Terms: $O$-terms with explicit constant + validity point, e.g., $B_{n\geq 100}(42 n^3)$
• Package extends SageMath; assume $n \to \infty$:
  • Monomially bounded variable $n^{\alpha} \leq k \leq n^{\beta}$
  • Taylor expansions with (and at) $B$-terms
• Installation:
  sage -pip install dependent_bterms
• Try it: behackl.dev/asy/dbt or static version

Creating an (enhanced) AsymptoticRing

			import dependent_bterms as dbt

			AR, n, k = dbt.AsymptoticRingWithDependentVariable(
			    'n^QQ',  # growth group, governs summand growth
			    'k', 0, 4/7,  # n^0 <= k <= n^(4/7)
			    bterm_round_to=3,  # number of decimal places kept when rounding
			    default_prec=5  # default precision for automatic expansions
			)
			

• Summands are ordered w.r.t. largest potential growth:

BTerm Arithmetic with a dependent variable

\[ \scriptsize \frac{1 - 2k + 3k^2 - 4k^3}{n^5} + B_{n\geq 10}(3 k^2 / n^3) =\, ? \]

• $k^3 n^{-5} = O(n^{-23/7}) = O(n^{-3})$ → absorbable!
• Crude estimate: $\scriptsize |1 - 2k + 3k^2 - 4k^3| \leq (1 + 2 + 3 + 4)k^3 = 10k^3$
• Use given bounds $k \leq n^{4/7}$ and $n\geq 10$:
  $\frac{10 k^3}{n^5} \leq \frac{10 k^2}{n^{31/7} } \leq \frac{10 k^2}{10^{10/7} n^3}$

Simplifications and Taylor Expansions

Act III — Towards an Explicit Error

• Fix $N = 10000$, consider growth of $F(n)$ for $n \geq N$
• Check inequality $F(n) \lt 0$ for $5 \leq n \lt N$ →

Tails: Identify range of negligible contributions

• Normalize by central binomial coefficient, want to work with $\binom{2n}{n-k} / \binom{2n}{n}$ instead
• Split sum over $1 \leq k\leq n$ into different regions:
  • $k \gt n/2$: ratio of binomial coefficients is exponentially small: $\scriptsize B_{n\geq N}(52/25\, e^{-n/4} n^7 \log\log n )$
  • $n^{\alpha} \leq k \leq n/2$ with $\alpha = 7/10$, sum bound: $\scriptsize 208/25 \log\log n \sum k^6 e^{-k^2 / n} $, then use integral estimate: $\scriptsize B_{n\geq N}(25073/5000\, e^{-n^{2/5} } n^{9/2} \log\log n)$
  

Dealing with Summands for $1\leq k \lt n^{\alpha}$

• Determine expansion of \[ \binom{2n}{n-k} / \binom{2n}{n} = e^{-k^2/n} \Big( 1 + \frac{k^2}{n^2} - \frac{k^4 + k^2}{6 n^3} + ...\Big) \]
• "Error part" contribution: $\scriptsize B_{n\geq N} (14671.8899\, \sqrt{n} \log\log n)$
• "Exact part": tail completion ↝ (sub)exponentially decaying errors \[ B_{n\geq N}\Big(2.6\, e^{-n^{2/5} } n^{19/4} \log\log n + 0.002\, e^{-n^{1/2} } n^{11/2} \Big) \]

Extract Asymptotics!

• Compute Mellin transform; converse mapping → shift line of integration, collect residues...
• Main term (with heavy cancellations): $-n^2/8 + n/24$
• Estimate shifted integrals:
  • $\Gamma$ causes fast decay, crude estimate for $\scriptsize|\operatorname{Im}(t)| \gt 100$
  • Numerical evaluation w/ interval arithmetic for $\scriptsize|\operatorname{Im}(t)| \leq 100$
  Error: $\scriptsize B_{n\geq N}(4065.31\, n^{3/4})$

Quantified Binomial Sum Asymptotics

After collecting and combining all error terms...