github.com/captn3m0/codechef.git

---
{"category_name":"medium","problem_code":"ARRPART","problem_name":"Array Partition","problemComponents":{"constraints":"- $1 \\leq T \\leq 7\\cdot 10^4$\n- $1 \\leq N \\leq 10^6$\n- $1 \\leq X \\leq 10^9$\n- $1 \\leq A_i \\leq 10^9$\n- The sum of $N$ across all test cases does not exceed $10^6$","constraintsState":true,"subtasks":"**Subtask #1 (20 points):** \n- $1 \\le T \\le 200$\n- $1 \\leq N \\leq 10^3$\n- The sum of $N$ across all test cases does not exceed $2\\cdot 10^3$\n- Time Limit = $1$ second\n\n**Subtask #2 (80 points):**  \n- Original constraints\n- Time Limit = $1.75$ seconds\n","subtasksState":true,"inputFormat":"- The first line of input contains a single integer $T$, denoting the number of test cases. The description of $T$ test cases follows.\n- Each test case contains two lines of input.\n- The first line of each test case consists of two space-separated integers $N$ and $X$.\n- The second line consists of $N$ space-separated integers denoting the elements of the array $A$.\n","inputFormatState":true,"outputFormat":"For each test case, output a single line containing $N$ space-separated integers, the $i$-th of which denotes the number of ways to partition the array into **exactly** $i$ non-empty subarrays such that the maxima of each of the $i$ subarrays are **at least** $X$. Print the answer modulo $998244353$.\n","outputFormatState":true,"sampleTestCases":{"0":{"id":1,"input":"3\n5 3\n4 1 7 1 6\n4 2\n2 2 2 1\n3 4\n5 6 7","output":"1 4 4 0 0 \n1 2 1 0 \n1 2 1","explanation":"In the below explanation, $[L, R]$ denotes the subarray consisting of elements $A_L, A_{L+1}, \\ldots, A_R$.\n\n**Test case $1$:**\n- The number of partitions having exactly $1$ subarray is $1$, which is the array itself.\n- The number of partitions having exactly $2$ subarrays is $4$, which are $[1,1]$+$[2,5]$, \n  $[1,2]$+$[3,5]$, $[1,3]$+$[4,5]$, $[1,4]$+$[5,5]$.\n- The number of partitions having exactly $3$ subarrays is $4$, which are $[1,1]$+$[2,3]$+$[4,5]$, \n  $[1,1]$+$[2,4]$+$[5,5]$, $[1,2]$+$[3,3]$+$[4,5]$, $[1,2]$+$[3,4]$+$[5,5]$.\n- The number of partitions having $4$ or $5$ subarrays is $0$.\n\n**Test case $2$:**\n- The number of partitions having exactly $1$ subarray is $1$, which is the array itself.\n- The number of partitions having exactly $2$ subarrays is $2$, which are $[1,1]$+$[2,4]$, \n  $[1,2]$+$[3,4]$.\n- The number of partitions having exactly $3$ subarrays is $1$, which is $[1,1]$+$[2,2]$+$[3,4]$.\n- The number of partitions having exactly $4$ subarrays is $0$.\n\n**Test Case $3$:**\n- The number of partitions having exactly $1$ subarray is $1$, which is the array itself.\n- The number of partitions having exactly $2$ subarrays is $2$, which are $[1,1]$+$[2,3]$, \n  $[1,2]$+$[3,3]$.\n- The number of partitions having exactly $3$ subarrays is $1$, which is $[1,1]$+$[2,2]$+$[3,3]$.","isDeleted":false}}},"video_editorial_url":"https://youtu.be/fjGMuN3HsN4","languages_supported":{"0":"CPP14","1":"C","2":"JAVA","3":"PYTH 3.6","4":"CPP17","5":"PYTH","6":"PYP3","7":"CS2","8":"ADA","9":"PYPY","10":"TEXT","11":"PAS fpc","12":"NODEJS","13":"RUBY","14":"PHP","15":"GO","16":"HASK","17":"TCL","18":"PERL","19":"SCALA","20":"LUA","21":"kotlin","22":"BASH","23":"JS","24":"LISP sbcl","25":"rust","26":"PAS gpc","27":"BF","28":"CLOJ","29":"R","30":"D","31":"CAML","32":"FORT","33":"ASM","34":"swift","35":"FS","36":"WSPC","37":"LISP clisp","38":"SQL","39":"SCM guile","40":"PERL6","41":"ERL","42":"CLPS","43":"ICK","44":"NICE","45":"PRLG","46":"ICON","47":"COB","48":"SCM chicken","49":"PIKE","50":"SCM qobi","51":"ST","52":"SQLQ","53":"NEM"},"max_timelimit":"1 - 1.75","source_sizelimit":50000,"problem_author":"kartikeya22022","problem_tester":"","date_added":"3-01-2022","tags":{"0":"kartikeya22022"},"problem_difficulty_level":"Medium","best_tag":"","editorial_url":"","time":{"view_start_date":1642411800,"submit_start_date":1642411800,"visible_start_date":1642411800,"end_date":1735669800},"is_direct_submittable":false,"problemDiscussURL":"https://discuss.codechef.com/search?q=ARRPART","is_proctored":false,"visitedContests":{},"layout":"problem"}
---
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Tracy is teaching Charlie maths via a game called $N$-Cube, which involves three sections involving $N$.

Tracy gives Charlie a number $N$, and Charlie makes a list of $N$-th powers of integers in increasing order $1^N, 2^N, 3^N, \dot, \text{so on}$. This teaches him exponentiation.

Then Charlie performs the following subtraction game $N$ times: Take all pairs of consecutive numbers in the list and take their difference. These differences then form the new list for the next iteration of the game. Eg, if $N$ was 6, the list proceeds as $[1, 64, 729, 4096 ... ]$ to $[63, 685, 3367 ...]$, and so on $5$ more times.

After the subtraction game, Charlie has to correctly tell Tracy the $N$-th element of the list. This number is the *value of the game*.

After practice Charlie became an expert in the game. To challenge him more, Tracy will give two numbers $M$ (where $M$ is a prime) and $R$ instead of just a single number $N$, and the game must start from $M_R - 1$ instead of $N$. Since the *value of the game* can now become large, Charlie just have to tell the largest integer $K$ such that $M_K$ divides this number. Since even $K$ can be large, output $K$ modulo 1000000007 ($10^9 + 7$).

<aside style='background: #f8f8f8;padding: 10px 15px;'><div>All submissions for this problem are available.</div></aside>