Conjecture on Supersequence Lower Bound related to Connell Sequence

TL;DR

The paper proves that the minimum length of a supersequence over a set of $8$ letters is $|\sigma|=52$, disproving the conjectured lower bound given by the partial sums of the geometric Connell sequence (which would suggest $51$). It introduces segmentation and reverse segmentation, along with a head-terminal-letter removal operation, to analyze internal letter frequencies and to reduce the problem to smaller alphabets. Through a sequence of technical lemmas and Claims, it derives tight lower bounds on letter frequencies ($f(\sigma,\mathbf8)\ge 5$, etc.) and shows that all viable configurations force the total length to $52$. This approach clarifies the intricacies of the general lower bound problem and provides a framework that could guide future tight bounds for larger finite sets, potentially bridging toward asymptotic optimality.

Abstract

This paper proves the minimum size of a supersequence over a set of eight elements is 52. This disproves a conjecture that the lower bound of the supersequence is the partial sum of the geometric Connell sequence. By studying the internal distribution of individual elements within sub-strings of the supersequence called segments, the proof provides important results on the internal structure that could help to understand the general lower bound problem for finite sets.

Theorems & Definitions (68)

• Definition 1
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• Definition 3
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 4
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• Definition 6