  A permutation is called {\it {block-wise simple}} if it contains no interval
of the form $p_1\oplus p_2$ or $p_1 \ominus p_2$. We present this new set of
permutations and explore some of its combinatorial properties. We present a
generating function for this set, as well as a recursive formula for counting
block-wise simple permutations. Following Tenner, who founded the notion of
interval posets, we characterize and count the interval posets corresponding to
block-wise simple permutations. We also present a bijection between these
interval posets and certain tiling's of the $n$-gon. Finally, we prove that the
bi-variate distribution of the descent and inverse descent numbers are
gamma-positive, provided the correctness of our recent conjecture on simple
permutations.
