On Counting Subsequences and Higher-Order Fibonacci Numbers

TL;DR

This paper explores the combinatorial problem of counting DNA strand sets with bounded common supersequence lengths, revealing a connection to higher-order Fibonacci numbers and their partial sums.

Contribution

It introduces a novel analysis linking DNA synthesis subsequence counts to higher-order Fibonacci numbers, advancing understanding of sequence combinatorics.

Findings

Number of subsequences relates to fourth-order Fibonacci partial sums

Provides formulas for counting DNA strand sets with bounded supersequence length

Establishes a new connection between DNA synthesis and Fibonacci number theory

Abstract

In array-based DNA synthesis, multiple strands of DNA are synthesized in parallel to reduce the time cost from the sum of their lengths to the length their shortest common supersequences. To maximize the amount of information that can be synthesized into DNA within a finite amount of time, we study the number of unordered sets of $n$ strands of DNA that have a common supersequence whose length is at most $t$. Our analysis stems from the following connection: The number of subsequences of A C G T A C G T A C G T ... is the partial sum (prefix sum) of the fourth-order Fibonacci numbers.