Binary strings of length $n$ with $x$ zeros and longest $k$-runs of zeros

TL;DR

This paper investigates the enumeration of binary strings with specific zero counts and longest zero runs, deriving recurrence relations and connecting to known combinatorial sequences, including palindromic variants.

Contribution

It introduces a recurrence relation for counting such binary strings and links these counts to well-known sequences like Fibonacci and compositions.

Findings

Derived recurrence relation for F_{n}(x,k)

Connected counts to Fibonacci, triangular numbers, and compositions

Extended results to palindromic binary strings

Abstract

In this paper, we study $F_{n}(x,k)$, the number of binary strings of length $n$ containing $x$ zeros and a longest subword of $k$ zeros. A recurrence relation for $F_{n}(x,k)$ is derived. We expressed few known numbers like Fibonacci, triangular, number of binary strings of length $n$ without $r$-runs of ones and number of compositions of $n+1$ with largest summand $k+1$ in terms of $F_{n}(x,k).$ Similar results and applications were obtained for \^F$_{n}(x,k),$ the number of all palindromic binary strings of length $n$ containing $x$ zeros and longest $k$-runs of zeros.