On the decomposition of $k$-noncrossing RNA structures

TL;DR

This paper analyzes the limit distribution of irreducible substructures in $k$-noncrossing RNA structures, revealing a Gamma distribution that characterizes their asymptotic behavior.

Contribution

It establishes the limit distribution of irreducible substructures in $k$-noncrossing RNA structures using vacillating tableaux and Gamma distribution analysis.

Findings

Limit distribution is a Gamma distribution with specific parameters.

Asymptotic behavior of irreducible substructures is characterized.

Provides a mathematical framework for understanding RNA structure complexity.

Abstract

An $k$-noncrossing RNA structure can be identified with an $k$-noncrossing diagram over $[n]$, which in turn corresponds to a vacillating tableaux having at most $(k-1)$ rows. In this paper we derive the limit distribution of irreducible substructures via studying their corresponding vacillating tableaux. Our main result proves, that the limit distribution of the numbers of irreducible substructures in $k$-noncrossing, $\sigma$-canonical RNA structures is determined by the density function of a $\Gamma(-\ln\tau_k,2)$-distribution for some $\tau_k<1$.