Full Linearization of Ranking and Unranking of Catalan Cipher Vectors Using Catalan Triangle Abstraction

Abstract

This paper demonstrates how to abstract a version of the Catalan Triangle necessary to compute the rank value from a given Catalan Cipher Vector, which is a process called ranking, and the process of obtaining a Catalan Cipher Vector from a given rank value, which is a process called unranking. That version of the Catalan Triangle is called the Bottom Ones Catalan Triangle and is not required to be computed in its entirety for the purpose of ranking and unranking, but only elements of it that are required for the current computation. A formula is derived to compute each element of this triangle and it is demonstrated how the processes of both ranking and unranking are fully linear.