Revisiting catalan numbers: an AB2 polymerization perspective

Number sequences, like Fibonacci, Fermat, Markov, Euler, Bernoulli, etc., have been popular in exemplifying a variety of scientific phenomena. Here, we explore the Catalan numbers in the context of ABm step polymerisation and develop a framework to derive its alternative closed form expression. Our approach harnesses the concepts of combinatorics and graph theory, in conjunction with kinetics of AB2 polymerisation to obtain the chain length distribution that directly gives the closed-form expression of Catalan number expressed as a bivariate distribution function. Furthermore, we validate our expression by comparing first 5000 Catalan numbers obtained from its traditional closed-form. As an offshoot, we discuss "pathwidth", a construct used in graph theory, as a better metrics for describing topology of polymer chains. The framework developed in this work can be extended to ABm step polymerisation and thus, facilitates topological characterisation of hyperbranched polymers (HPs) that ultimately, dictates their structure-property relationships.