CF2222H.Counting Sort?

I want to end this life impulsively for the decisive time instead of living on the rails that have been laid out for me.

Kashii Moimi & Kaai Yuki, The Decisive Hour

For an array $ [b_1, b_2, \ldots, b_m] $ where $ 0 \le b_i \le m $ , define $ f(b) $ as an array $ [c_1, c_2, \ldots, c_m] $ such that $ c_i $ is the number of occurrences of $ i $ in the array $ b $ .

Define the value $ g(b) $ of an array $ [b_1, b_2, \ldots, b_m] $ , where $ 0 \le b_i \le m $ , as follows:

• Initially, there is an empty set $ S $ . Create an array $ d $ that is initially equal to $ b $ .
• Then, you perform the following operation $ 100^{100} $ times: Insert $ d $ into $ S $ , then set $ d \gets f(d) $ .
• $ g(b) $ is the number of distinct arrays in $ S $ after the operations.

Now you are given two integers $ n $ and $ k $ , as well as an array $ [r_1, r_2, \ldots, r_n] $ . For every $ 1\le p\le k $ , count the number of arrays $ [a_1, a_2, \ldots, a_n] $ where $ 0 \le a_i \le r_i $ such that $ g(a) = p $ . Since this number may be large, output it modulo $ 998,244,353 $ .

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 1000 $ ). The description of the test cases follows.

The first line of each test case contains two integers $ n $ and $ k $ ( $ 1 \le n \le 50 $ , $ 1 \le k \le 1000 $ ).

The second line contains $ n $ integers $ r_1, r_2, \ldots, r_n $ ( $ 0 \le r_i \le n $ ).

It is guaranteed that the sum of $ n^3 $ over all test cases does not exceed $ 50^3 $ .

For each test case, output $ k $ integers — the number of arrays $ [a_1, a_2, \ldots, a_n] $ with $ 0 \le a_i \le r_i $ such that $ g(a) = 1, 2, \ldots, k $ , respectively.

Print the answer modulo $ 998,244,353 $ .

In the first test case, $ g([0, 0]) = g([1, 0]) = 1 $ , $ g([0, 1]) = 2 $ , $ g([2, 0]) = g([0, 2]) = 3 $ , $ g([1, 1]) = g([2, 2]) = 4 $ , and $ g([1, 2]) = g([2, 1]) = 5 $ .

In the second test case, $ f([4, 5, 1, 4, 5]) = [1, 0, 0, 2, 2] $ , $ f([1, 0, 0, 2, 2]) = [1, 2, 0, 0, 0] $ , $ f([1, 2, 0, 0, 0]) = [1, 1, 0, 0, 0] $ , $ f([1, 1, 0, 0, 0]) = [2, 0, 0, 0, 0] $ , $ f([2, 0, 0, 0, 0]) = [0, 1, 0, 0, 0] $ , $ f([0, 1, 0, 0, 0]) = [1, 0, 0, 0, 0] $ , $ f([1, 0, 0, 0, 0]) = [1, 0, 0, 0, 0] $ . So, $ g([4, 5, 1, 4, 5]) = 7 $ .