Equivalently, it is the direct sum of decreasing permutations.
[1] One of the earlier works establishing the significance of layered permutations was Bóna (1999), which established the Stanley–Wilf conjecture for classes of permutations forbidding a layered permutation, before the conjecture was proven more generally.
That is, no three elements in the permutation (regardless of whether they are consecutive) have the same ordering as either of these forbidden triples.
A layered permutation on the numbers from
can be uniquely described by the subset of the numbers from
that are the first element in a reversed block.
is always the first element in its reversed block, so it is redundant for this description.)
For instance, the Gilbreath permutations are counted by the same function
[3] The shortest superpattern of the layered permutations of length
these numbers are and in general they are given by the formula Every layered permutation is an involution.
[5] The layered permutations are a subset of the stack-sortable permutations, which forbid the pattern 231 but not the pattern 312.
Like the stack-sortable permutations, they are also a subset of the separable permutations, the permutations formed by recursive combinations of direct and skew sums.