[title inspired by this blog]

Hello everyone,

today, during a NEERC virtual contest, I found an unintended solution for problem 1089I - Interval-Free Permutations. I've checked all the official submissions and no one of them uses my solution, so I think it's worth sharing it.

Abridged statement: count the permutations of $$$[1, \dots, n]$$$ such that there are no subarrays of length between $$$2$$$ and $$$n-1$$$ where all the values are contiguous. For example, the permutation $$$[2,8,4,6,3,5,1,7]$$$ is bad because it contains $$$[4,6,3,5]$$$ as a subarray. Output the answer (modulo a prime, given in the input) for all $$$1 \leq n \leq 400$$$.

My solution:

• Let's use PIE (inclusion-exclusion principle) on minimal bad subarrays.
• Let's use Connected Components DP, somehow keeping track of minimal bad subarrays.
  
  
• Let $$$dp_{i,j,k}$$$ be the number of ordered sets of $$$j$$$ connected components with total length $$$i$$$, and $$$k =$$$ parity of minimal bad subarrays. Then, the number of good permutations of length $$$i$$$ is $$$dp_{i,1,0} - dp_{i,1,1}$$$.
  Instead of adding elements one at a time to the permutation, let's consider two cases:
  - We add only one element (using the standard Connected Components DP transitions);
  - We add a minimal bad subarray of length $$$2 \leq l \leq i-1$$$ (the transitions are similar, but using $$$dp_{i-l,\text{*},k \oplus 1}$$$ instead of $$$dp_{i-1, \text{*}, k}$$$. Note that the number of ways to add a minimal bad subarray of length $$$l$$$ is equal to the number of good permutations of length $$$l$$$.
• When we calculate $$$dp_{i,*,*}$$$, we assume that $$$dp_{j,1,*} = 0$$$ ($$$j < i$$$), because the corresponding elements are good as arrays but bad as subarrays.

This solution is actually wrong: in most cases, it produces the correct output $$$\pm 2$$$! It turns out it's enough to add $$$-2 \cdot (-1)^n$$$ to the result, for $$$n \geq 3$$$. (AC code: 181878668)

So my questions are:

• Why is the initial solution wrong?
  

• Why is the solution with $$$-2 \cdot (-1)^n$$$ correct? Actually I don't know, I've just found the formula using the samples.
• Can this solution be generalized to solve harder problems? For example,
  "An array is weird if the local minimums are bitonic (i.e., decreasing, then increasing). Count the weird permutations of $$$[1, \dots, n]$$$ such that there are no weird subarrays of length between $$$2$$$ and $$$n-1$$$ where all the values are contiguous."