Decomposition of permutations

In this problem, a permutation of $n$ elements is defined as a sequence of integers of length $n$, where each number from $0$ to n-1 appears exactly once. Let the $i$-th element of the permutation $p$ be denoted as p_i, where $i$ is an index from $0$ to n-1. The product of two permutations $a$ and $b$ of $n$ elements is another permutation $c$ of $n$ elements, such that c_i = a_{b_i}. Note that permutation multiplication is left-associative, meaning operations are performed from left to right.

Among permutations, we focus on double permutations. A double permutation can be represented as a union of two non-overlapping increasing sequences. For example, the permutation (1, 3, 4, 0, 5, 2) is a double permutation because it can be split into two increasing sequences: (1, 3, 4, 5) and $(0,2)$. In contrast, the permutation (5, 1, 3, 4, 0, 2) is not a double permutation.

Your task is to write a program that decomposes a given permutation into a product of double permutations. The number of double permutations in the product should be as small as possible. Your program should find a solution consisting of no more than $15$ double permutations.