  Patterned random matrices such as the reverse circulant, the symmetric
circulant, the Toeplitz and the Hankel matrices and their almost sure limiting
spectral distribution (LSD), have attracted much attention. Under the
assumption that the entries are taken from an i.i.d. sequence with finite
variance, the LSD are tied together by a common thread -- the $2k$th moment of
the limit equals a weighted sum over different types of pair-partitions of the
set $\{1, 2, \ldots, 2k\}$ and are universal. Some results are also known for
the sparse case. In this paper we generalise these results by relaxing
significantly the i.i.d. assumption. For our models, the limits are defined via
a larger class of partitions and are also not universal. Several existing and
new results for patterned matrices, their band and sparse versions, as well as
for matrices with continuous and discrete variance profile follow as special
cases.
