Record values in appending and prepending bitstrings to runs of binary digits

TL;DR

This paper characterizes integers that set records when appending or prepending binary strings to runs of 1's and 0's, revealing that the record set is independent of the specific nonempty strings added, as long as they are of equal length.

Contribution

It provides a simple characterization of record-setting integers in sequences formed by adding binary strings to runs of 1's and 0's, independent of the specific strings used.

Findings

Record-setting integers are characterized by a simple property.

The set of record integers is independent of the added strings, given they are nonempty and of equal length.

The result applies to sequences formed by appending or prepending binary strings to runs of bits.

Abstract

In this short note, we show a simple characterization of integers that reach records for a sequence described by adding binary strings to runs of 1's and 0's in a binary representation. In particular, we show that this set does not depend on the added strings as long as they are nonempty and of the same length.