Chess Sliders

A Slider is a chess piece that can move one square left or right.
This problem uses a cylindrical chess board where the left hand edge of the board is connected to the right hand edge. This means that a Slider that is on the left hand edge of the chess board can move to the right hand edge of the same row and vice versa.
Let $L(N,K)$ be the number of ways $K$ non-attacking Sliders can be placed on an $N \times N$ cylindrical chess-board.
For example, $L(2,2)=4$ and $L(6,12)=4204761$.
Find $L(10^9,10^{15}) \bmod \left(10^7+19\right)^2$.

Let’s use the list slider problem as an example.

If we understand the slider chess piece correctly, the slider moves exactly one square left or right on each move, and the board is cylindrical in that the left hand edge of the board is connected to the right hand edge. This means a slider on the leftmost border of the board can appear on the rightmost border in its next move, and vice versa.

The problem boils down to how many possible arrangements there are of placing k non-attacking sliders on a cylindrical chess board of size NxN, where by “non-attacking” we mean no two sliders can occupy the same square.

Due to the size and scale of the numbers involved (N=10^9 and K=10^15), it’s clear that a direct combinatoric counting is impractical. Therefore, we can anticipate that such a problem is not intended to be solved exactly, but rather modulo a particular number for a practical computation. And this is indeed the case: the problem asks for L(N, K) mod (10^7+19)^2.

To solve this problem exactly would require a detailed understanding of advanced combinatorics, specifically combinatorics in non-Euclidean spaces, given that the chessboard is “cylindrical”. This also requires to deal with the large numbers problem, where exact calculations are not practical or feasible given current computational limitations. Standard combinatoric methods become unwieldy when dealing with combinations and permutations with such large numbers of items.

Although it may be too complicate to solve in the scope of this platform, it is important to know that in similar problems a solution would involve deep insights on the systematic arrangement of the non-attacking sliders using Combinatorics, and then determining the mathematical properties of these arrangements that allow us to calculate the total number of them modulo the given number. Calculations modulo a prime number p are common in number theory and algebraic number theory due to their nice properties. In Algebraic Number Theory, the Cyclotomic field Q(ξ_p) is often considered where ξ_p is a primitive p-th root of unity and p is prime.

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