Some Notes on Pairs in Binary Strings
Jeremy M. Dover

TL;DR
This paper derives a closed-form formula for counting binary strings of length n with specified numbers of consecutive 0s and 1s, considering the string as cyclic, and explores related combinatorial sequences.
Contribution
It introduces a closed-form solution for a problem posed on Math Stack Exchange, linking it to other combinatorial sequences and extending previous results.
Findings
Derived a closed-form expression for the count of binary strings with given consecutive pairs
Connected the problem to known combinatorial sequences
Provided solutions considering the string as cyclic
Abstract
Seth (Mathematics Stack Exchange, http://math.stackexchange.com/q/1812699) posed a problem that is equivalent to the following: how many binary strings of length n have exactly k pairs of consecutive 0s and exactly m pairs of consecutive 1s, where the first and last bits are considered as being consecutive? In this paper, we provide a closed form solution which also solves a related problem with some interesting connections to other combinatorial sequences.
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Taxonomy
Topicsgraph theory and CDMA systems · semigroups and automata theory · Coding theory and cryptography
