Blockwise simple permutations

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.