Some Notes on Pairs in Binary Strings

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.