  The novelty of this paper is to construct the explicit combinatorial formula
for the number of all distinct fuzzy matrices of finite order, which leads us
to invent a new sequence. In order to achieve this new sequence, we analyze the
behavioral study of equivalence classes on the set of all fuzzy matrices of a
given order under a suitable natural equivalence relation. In addition this
paper characterizes the properties of non-equivalent classes of fuzzy matrices
of order n with elements having degrees of membership values anywhere in the
closed unit interval [0,1]. Further, this paper also derives some important
relevant results by enumerating the number of all distinct fuzzy matrices of a
given order in general. And also, we achieve these results by incorporating the
notion of k-level fuzzy matrices, chains, and flags (maximal chains).
  Keywords: Fuzzy matrices; k-level fuzzy matrices; Chains; Flags; Binomial
numbers
