  In this paper, we consider the number of occurrences of descents, ascents,
123-subwords, 321-subwords, peaks and valleys in flattened permutations, which
were recently introduced by Callan in his study of finite set partitions. For
descents and ascents, we make use of the kernel method and obtain an explicit
formula (in terms of Eulerian polynomials) for the distribution on
$\mathcal{S}_n$ in the flattened sense. For the other four patterns in
question, we develop a unified approach to obtain explicit formulas for the
comparable distributions. We find that the formulas so obtained for 123- and
321-subwords can be expressed in terms of the Chebyshev polynomials of the
second kind, while those for peaks and valleys are more related to the Eulerian
polynomials. We also provide a bijection showing the equidistribution of
descents in flattened permutations of a given length with big descents in
permutations of the same length in the usual sense.
