Problem 981

The Quaternion Group II

Starting from an empty string, we want to build a string with letters “x”, “y”, “z”. At each step, one of the following operations is performed:

insert two consecutive identical letters “xx”, “yy” or “zz” anywhere into the string;
replace one letter in the string with two consecutive letters, according to the rule: “x” $\to$ “yz”, “y” $\to$ “zx”, “z” $\to$ “xy”;
exchange two consecutive different letters in the string, e.g. “xy” $\to$ “yx”, “zx” $\to$ “xz”, etc.

A string is called neutral if it is possible to produce the string from the empty string after an even number of steps.

Let $N(X, Y, Z)$ be the number of neutral strings which contain $X$ copies of “x”, $Y$ copies of “y” and $Z$ copies of “z”.

For example, $N(2, 2, 2) = 42$ and $N(8, 8, 8) = 4732773210$.

Find the sum of $N(i^3, j^3, k^3)$ for $0 \le i, j, k \lt 88$. Give your answer modulo $888\ 888\ 883$.

四元数群（二）

从空字符串开始，我们逐步构造一个仅由字母”x”、”y”、”z”构成的新字符串，其中每一步执行以下操作之一：

在字符串的任意位置插入两个连续的相同字母”xx”、”yy”或”zz”；
将字符串中的一个字母替换为两个连续字母，规则如下：”x” $\to$ “yz”，”y” $\to$ “zx”，”z” $\to$ “xy”；
交换字符串中两个相邻的不同字母，例如”xy” $\to$ “yx”，”zx” $\to$ “xz”，等等。

如果一个字符串可以从空字符串经过偶数步操作生成，则称之为中性字符串。

记$N(X, Y, Z)$为包含$X$个”x”、$Y$个”y”和$Z$个”z”的中性字符串数目。

例如，$N(2, 2, 2) = 42$，$N(8, 8, 8) = 4732773210$。

求所有满足$0 \le i, j, k \lt 88$的$N(i^3, j^3, k^3)$之和，并对$888\ 888\ 883$取余作为你的答案。