Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Binary strings of length $n$ with $x$ zeros and longest $k$-runs of
  zeros   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.

32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess