Seating Arrangement Approach

The new catchphrase of Jumbo Airlines is "No annoying neighbors, each flight a unique experience!"

And as in most cases, the advertisement was produced by the marketing department, without ever consulting the engineers. They only learned about it after the boss asked them to "handle it ASAP".

There are M seats in each row, and there are N rows of seats in the airplane. Hence the seats form an M * N grid. (For the purpose of this problem we will ignore the presence of aisles.) The airline sells exactly K tickets for each flight.

To make sure that the "no annoying neighbors" part of the motto is satisfied, the seating must obey the following rule: Whenever a seat is occupied, the seats immediately in front of it and behind it, as well as the seats immediately to the left and to the right must remain free.

An allowed arrangement is a set of K occupied seats that obeys the rule above.

The "unique experience" part of the motto is then satisfied by using a different arrangement of occupied seats for each flight. (Two seating arrangements are different if there is at least one seat which is occupied in one arrangement and free in the other.)

Problem specification You are given the numbers M, N and K. Find the number of different allowed seating arrangements.

As this number can be very large, we're only interested in its value modulo 420047.

This is an intriguing problem that involves dynamic programming. To start, let's break down the problem into simpler components that we can handle step by step. Here are a couple of questions to consider:

1. Basic Understanding: Do you understand the constraints of the seating arrangement? Specifically, the rule that if a seat is occupied, the seats to its left, right, front, and back must remain empty. How does this affect the arrangement of occupied seats?

2. Initial Thoughts on Approach: Given the constraints, how would you begin to think about organizing the seats? One way to think about it is in terms of choosing a seat to occupy and then seeing how that affects the options for the remaining seats.

Considering these points, can you start to formulate an initial approach to the problem? Remember, in dynamic programming, we try to break down a problem into smaller sub-problems and then build up the solution. How could we apply this here?

I understand the constraints. I would want to somehow break down the grid into smaller grids but the solution would depend on if the seats at the edge of the smaller sub grid was occupied or unoccupied. Might be a bit mask DP not sure. For every seat it can be occupied or unoccupied so you need to count both.

Great! You're on the right track with considering breaking down the grid and the idea of using a bitmask for the DP state. Let's refine this approach:

1. Sub-grid Analysis: Consider a row in the airplane. Due to the constraint that an occupied seat must have adjacent seats (left and right) empty, we can represent the state of each row with a bitmask. For instance, if there are M seats in a row, a bitmask of M bits can represent which seats are occupied (1) and which are unoccupied (0).

2. State Transition: The challenge is to figure out how the state of one row affects the possible states of the next row, given that seats in consecutive rows also cannot be adjacent to each other (vertically). This means that if a seat is occupied in one row, the seats directly in front and behind in adjacent rows must be unoccupied.

3. Dynamic Programming State: We can define our DP state as DP[i][mask], which represents the number of ways to arrange passengers in the first i rows with the i-th row having an occupancy state represented by mask.

4. Transition Between States: The key part is to determine valid transitions from DP[i][mask1] to DP[i+1][mask2], ensuring that no two passengers violate the seating constraints.

5. Base Cases and Initialization: The base case would be the first row, where any arrangement of K seats that don't have adjacent occupied seats is valid.

6. Result: The total number of arrangements will be the sum of all DP[N][mask] where mask represents any valid arrangement of the last row.

With this outline, let's dive into the first Socratic question:

• How do you think we can generate valid transitions between states? Consider how a given arrangement (mask) in one row restricts or allows certain arrangements in the adjacent row.

Thinking through this will help in formulating the recurrence relation for the DP states.