Decomposing Catalan number interpretations in lattice path counting with applications to Caribbean hurricane intensification

Introduction

From a regional perspective, the Caribbean basin generates roughly 12% of global tropical cyclone activity but accounts for a disproportionate share of rapid-intensification events episodes where maximum sustained wind speed increases by 30 knots or more within 24 hours [1] . Forecasting rapid intensification remains one of the hardest problems in operational meteorology because it depends on nonlinear interactions between oceanic heat content, atmospheric shear, and internal storm dynamics [2] . Standard intensity models use physical predictors (SST, shear, moisture), but the nonlinearity of the intensification process suggests that structural complexity measures might add predictive value. Catalan numbers arise wherever recursive combinatorial structures are counted binary trees, Dyck paths, polygon triangulations, ballot sequences, and stack-sortable permutations are all counted by the same sequence 1, 1, 2, 5, 14, 42, [3] . Each interpretation encodes a different aspect of structural complexity, and this research asks whether these aspects map onto distinct features of hurricane intensification. The approach decomposes the intensification trajectory into a lattice path on a time-pressure grid and computes Catalan indices for each of the five interpretations [4] .

Sensitivity analysis

The Catalan index was recomputed under three perturbation regimes: (i) adding Gaussian noise (SD = 5 knots) to the intensity record, (ii) shifting the rapid-intensification threshold from 30 to 25 and 35 knots, and (iii) varying the lattice discretisation from 12-hourly to 6https://www.mathsjournal.com Hourly steps. The Dyck-path correlation remained above 0.65 under all perturbations, with the strongest sensitivity to the threshold shift (rho dropped from Lattice resolution had minimal effect (rho = 0.71 at 6-hourly).

Scenario development

Two SST scenarios were considered. The baseline scenario used observed SST from ERA5 reanalysis. The warmanomaly scenario added a uniform +1 C perturbation, consistent with projected mid-century Caribbean warming [5] . Under the warm scenario, the counterfactual analysis attributed 14 additional rapid-intensification events (from 87 to 101) to the SST increase, and the mean Dyck-path complexity index rose from 4.3 to 5.1.

Material and methods

Material

The International Best Track Archive for Climate Stewardship (IBTrACS) provided 6-hourly intensity records for 312 Caribbean tropical cyclones from 1980 to 2023. Rapid intensification was defined as a wind-speed increase of >= 30 knots in 24 hours. SST fields were obtained from ERA5 at 0.25-degree resolution and kriging-interpolated to stormcentre tracks using a Matern covariance with range 200 km. All modelling was performed at the Udaipur Department of Algebraic Studies between January 2023 and September 2023.

Methods

Each storm's intensity trajectory was discretised onto a lattice with horizontal axis = time (6-hourly steps) and vertical axis = intensity (5-knot bins). Five Catalan indices were computed: (1) Dyck-path count = number of sub-paths that touched but never crossed the diagonal; (2) ballot-sequence length = longest initial segment satisfying the ballot condition; (3) binary-tree depth = maximum depth of the recursive decomposition tree; (4) triangulation count = number of non-crossing partition elements; (5) stack-sortable permutation rank [3] . Spearman rank correlations between each index and the rapid-intensification binary outcome were computed. The game-theoretic model treated the hurricane as a player maximizing kinetic energy against an environment player controlling shear, with Nash equilibrium intensity as the predicted outcome [6] . Counterfactual intensification was estimated by re-running the game under altered SST [7] . Basis expansion represented the Catalan generating function C(x) = (1-sqrt (1-4x))/ (2x) as a polynomial in SST anomaly to create a hybrid forecast model [4] . Dyck-path complexity showed the strongest correlation with rapid intensification (rho = 0.72). All five Catalan indices were significantly higher in rapid-intensification storms, confirming that combinatorial complexity captures structural features of the intensification trajectory. The heatmap shows that Dyck paths and ballot sequences had the strongest correlations with Categories 4-5, while stacksortable permutations showed a more uniform distribution across categories.

Results

Discussion

The Dyck-path interpretation's superiority likely reflects its sensitivity to the 'touching-without-crossing' property of the intensity trajectory: rapid-intensification storms repeatedly approach but do not cross a critical intensity threshold before breaking through, mirroring the ballot-problem dynamics encoded in Dyck paths [3] . The game-theoretic model's Nash equilibrium provided a complementary physical interpretation: the hurricane's optimal strategy against environmental shear involves accumulating potential intensity through repeated near-threshold excursions, a pattern that increases Dyck-path complexity [6] . The counterfactual analysis confirmed that SST modulates this complexity, with warmer waters allowing more near-threshold excursions before breakthrough [7] . A limitation is that the lattice discretisation imposes an artificial resolution on the continuous intensity trajectory. Finer discretisation preserves more structure but inflates the Catalan indices, requiring normalisation. Future work could use continuous analogues of Catalan structures (e.g., Brownian excursion counts) to avoid discretisation artifacts [8] .

Conclusion

This research decomposed five Catalan number interpretations in lattice path counting and correlated them with Caribbean hurricane rapid intensification. Dyck-path complexity showed the strongest association (rho = 0.72), and counterfactual analysis attributed 14 additional rapidintensification events to a +1 C SST anomaly. Game-theoretic modelling provided a physical interpretation of the combinatorial complexity. These findings introduce combinatorial diagnostics as a novel tool for tropical cyclone research.

Acknowledgement

Acknowledgements Funding sourcesInstitutional funds from the Udaipur Department of Algebraic Studies. Institutional supportThe authors thank NOAA for IBTrACS data and the Udaipur Department for computing resources. Contributions not qualifying for authorshipData extraction was assisted by postgraduate students.

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