On the Partial Sum of Subword-Counting Sequences

TL;DR

This paper investigates the behavior of sums involving the number of scattered subsequence occurrences of binary words, identifying classes of words with sums that grow slower than linearly, revealing new combinatorial properties.

Contribution

It characterizes classes of binary words for which the partial sums of alternating counts of scattered subsequences grow sublinearly, advancing understanding of binary word combinatorics.

Findings

Identified classes of words with sublinear partial sum growth

Provided bounds for sums involving scattered subsequences

Enhanced understanding of binary word combinatorics

Abstract

Let $w$ be a finite word over the alphabet $\{0,1\}$. For any natural number $n$, let $s_w(n)$ denote the number of occurrence of $w$ in the binary expansion of $n$ as a scattered subsequence. We study the behavior of the partial sum $\sum_{n=0}^N(-1)^{s_w(n)}$ and characterize several classes of words $w$ satisfying $\sum_{n=0}^N(-1)^{s_w(n)}= O(N^{1-\epsilon})$ for some $\epsilon >0$.