Maximum Distance PermutationsDefine the distance between 22 permutations†† 𝐴A and 𝐵B, each of length 𝑁N, as the minimum value of ∣𝐴𝑖−𝐵𝑖∣‡∣A i​ −B i​ ∣ ‡ for 1≤𝑖≤𝑁1≤i≤N.Find a pair of permutations of length 𝑁N which have the maximum possible distance among all pairs of permutations of length 𝑁N.†† A permutation of length 𝑁N is an array of 𝑁N distinct elements which are all in the range [1,𝑁][1,N]. For example, [2,3,1,4][2,3,1,4] is a permutation of length 44 whereas [2,4,3][2,4,3] and [1,1,2][1,1,2] are not.‡‡ ∣𝑋∣∣X∣ denotes the absolute value of 𝑋X. For example, ∣−5∣=∣5∣=5∣−5∣=∣5∣=5.Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case consists of a single integer 𝑁N - the length of the permutations.Output FormatFor each test case, output 22 lines.The first line containing 𝑁N space-separated integers - 𝐴1,𝐴2,𝐴3,…,𝐴𝑁A 1​ ,A 2​ ,A 3​ ,…,A N​ , denoting the permutation 𝐴A.The second line containing 𝑁N space-separated integers - 𝐵1,𝐵2,𝐵3,…,𝐵𝑁B 1​ ,B 2​ ,B 3​ ,…,B N​ , denoting the permutation 𝐵B.The permutations 𝐴A and 𝐵B must have the maximum distance possible. If multiple answers are possible, all will be accepted.Constraints1≤𝑇≤1001≤T≤1001≤𝑁≤1001≤N≤100Sample 1:InputOutput3123111 22 11 2 33 1 2Explanation:Test case 1 : There is only one possible permutation of length 𝑁N, and thus only one possible pair, which has distance 00.Test case 2 : The permutations [1,2][1,2] and [2,1][2,1] have a distance of 11. This can be shown to be the maximum possible distance.

For a permutation 𝑃P of size 𝑁N, and an integer 𝑘 (1≤𝑘≤𝑁)k (1≤k≤N), define 𝑓(𝑃,𝑘)f(P,k) as ∑𝑖=1𝑁−𝑘+1max⁡(𝑃𝑖,𝑃𝑖+1,𝑃𝑖+2,…,𝑃𝑖+𝑘−1)i=1∑N−k+1​ max(P i​ ,P i+1​ ,P i+2​ ,…,P i+k−1​ ), i.e. the sum of maximum elements in all 𝑘k - sized subarrays.Let 𝑔(𝑁,𝑘)g(N,k) denote the number of permutations 𝑃P of size 𝑁N which have the maximum value of 𝑓(𝑃,𝑘)f(P,k).Given an integer 𝑁N, find the value of 𝑔(𝑁,𝑘)g(N,k) for each 1≤𝑘≤𝑁1≤k≤N.Since the value can be huge, print it modulo 109+710 9 +7.Note that a permutation of size 𝑁N consists of all elements from 11 to 𝑁N exactly once.Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case consists of a single integer 𝑁N - the size of the permutation.Output FormatFor each test case, output on a new line, 𝑁N space-separated integers denoting the value of 𝑔(𝑁,𝑘)g(N,k) for each 1≤𝑘≤𝑁1≤k≤N, modulo 109+710 9 +7.Constraints1≤𝑇≤5001≤T≤5001≤𝑁≤3⋅1051≤N≤3⋅10 5 The sum of 𝑁N over all test cases does not exceed 3⋅1053⋅10 5 .Sample 1:InputOutput312312 26 2 6Explanation:Test case 11: There is only one possible permutation of size 11. Thus, 𝑓([1],1)=1f([1],1)=1 and 𝑔(1,1)=1g(1,1)=1.Test case 22:𝑔(2,1)=2g(2,1)=2 as there are two permutations with maximum 𝑓(𝑃,1)f(P,1)𝑓([1,2],1)=1+2=3f([1,2],1)=1+2=3𝑓([2,1],1)=2+1=3f([2,1],1)=2+1=3𝑔(2,2)=2g(2,2)=2 as there are two permutations with maximum 𝑓(𝑃,2)f(P,2)𝑓([1,2],2)=2f([1,2],2)=2𝑓([2,1],2)=2f([2,1],2)=2Test case 33: For 𝑘=1k=1 and 𝑘=3k=3, all permutations have the same value of 𝑓(𝑃,𝑘)f(P,k), thus the answer is 66. However, for 𝑘=2k=2, the optimal permutations are [2,3,1][2,3,1] and [1,3,2][1,3,2] only.

Make PermutationChef has an array 𝐴A of size 𝑁N. He wants to make a permutation†† using this array.Find whether there exists an array 𝐵B consisting of 𝑁N non-negative integers, such that the array 𝐶C constructed as 𝐶𝑖=𝐴𝑖+𝐵𝑖C i​ =A i​ +B i​ is a permutation.†† A permutation of size 𝑁N is an array of 𝑁N distinct elements in the range [1,𝑁][1,N]. For example, [4,2,1,3][4,2,1,3] is a permutation of size 44, while [3,2,2,1][3,2,2,1] and [1,3,4][1,3,4] are not.Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case consists of multiple lines of input.The first line of each test case consists of 𝑁N - the size of the array 𝐴AThe next line contains 𝑁N space-separated integers - 𝐴1,𝐴2,…,𝐴𝑁A 1​ ,A 2​ ,…,A N​ - the elements of array 𝐴A.Output FormatFor each test case, output YES if there exists an array 𝐵B such that array 𝐶C constructed as 𝐶𝑖=𝐴𝑖+𝐵𝑖C i​ =A i​ +B i​ is a permutation, otherwise output NO.You may print each character of the string in uppercase or lowercase (for example, the strings YES, yEs, yes, and yeS will all be treated as identical).Constraints1≤𝑇≤1001≤T≤1001≤𝑁≤1001≤N≤1001≤𝐴𝑖≤𝑁1≤A i​ ≤NSample 1:InputOutput454 1 3 2 152 4 3 4 21161 1 1 1 6 6YESNOYESNOExplanation:Test case 11 : Consider 𝐵=[0,4,0,0,0]B=[0,4,0,0,0]. The corresponding array 𝐶C becomes [4,5,3,2,1][4,5,3,2,1], which is a permutation. Some other possible values of 𝐵B for which 𝐶C is a permutation are [1,0,1,1,1][1,0,1,1,1] and [1,3,0,0,0][1,3,0,0,0].Test case 22 : It can be proven that no valid array 𝐵B exists.

Initially, we had one array, which was a permutation of size n𝑛 (an array of size n𝑛 where each integer from 11 to n𝑛 appears exactly once).We performed q𝑞 operations. During the i𝑖-th operation, we did the following:choose any array we have with at least 22 elements;split it into two non-empty arrays (prefix and suffix);write two integers li𝑙𝑖 and ri𝑟𝑖, where li𝑙𝑖 is the maximum element in the left part which we get after the split, and ri𝑟𝑖 is the maximum element in the right part;remove the array we've chosen from the pool of arrays we can use, and add the two resulting parts into the pool.For example, suppose the initial array was [6,3,4,1,2,5][6,3,4,1,2,5], and we performed the following operations:choose the array [6,3,4,1,2,5][6,3,4,1,2,5] and split it into [6,3][6,3] and [4,1,2,5][4,1,2,5]. Then we write l1=6𝑙1=6 and r1=5𝑟1=5, and the arrays we have are [6,3][6,3] and [4,1,2,5][4,1,2,5];choose the array [4,1,2,5][4,1,2,5] and split it into [4,1,2][4,1,2] and [5][5]. Then we write l2=4𝑙2=4 and r2=5𝑟2=5, and the arrays we have are [6,3][6,3], [4,1,2][4,1,2] and [5][5];choose the array [4,1,2][4,1,2] and split it into [4][4] and [1,2][1,2]. Then we write l3=4𝑙3=4 and r3=2𝑟3=2, and the arrays we have are [6,3][6,3], [4][4], [1,2][1,2] and [5][5].You are given two integers n𝑛 and q𝑞, and two sequences [l1,l2,…,lq][𝑙1,𝑙2,…,𝑙𝑞] and [r1,r2,…,rq][𝑟1,𝑟2,…,𝑟𝑞]. A permutation of size n𝑛 is called valid if we can perform q𝑞 operations and produce the given sequences [l1,l2,…,lq][𝑙1,𝑙2,…,𝑙𝑞] and [r1,r2,…,rq][𝑟1,𝑟2,…,𝑟𝑞].Calculate the number of valid permutations.InputThe first line contains two integers n𝑛 and q𝑞 (1≤q<n≤3⋅1051≤𝑞<𝑛≤3⋅105).The second line contains q𝑞 integers l1,l2,…,lq𝑙1,𝑙2,…,𝑙𝑞 (1≤li≤n1≤𝑙𝑖≤𝑛).The third line contains q𝑞 integers r1,r2,…,rq𝑟1,𝑟2,…,𝑟𝑞 (1≤ri≤n1≤𝑟𝑖≤𝑛).Additional constraint on the input: there exists at least one permutation which can produce the given sequences [l1,l2,…,lq][𝑙1,𝑙2,…,𝑙𝑞] and [r1,r2,…,rq][𝑟1,𝑟2,…,𝑟𝑞].OutputPrint one integer — the number of valid permutations, taken modulo 998244353998244353.ExamplesinputCopy6 36 4 45 5 2outputCopy30inputCopy10 1109outputCopy1814400inputCopy4 124outputCopy8

A permutation of an array of integers is an arrangement of its members into a sequence or linear order.For example, for arr = [1,2,3], the following are all the permutations of arr: [1,2,3], [1,3,2], [2, 1, 3], [2, 3, 1], [3,1,2], [3,2,1].The next permutation of an array of integers is the next lexicographically greater permutation of its integer. More formally, if all the permutations of the array are sorted in one container according to their lexicographical order, then the next permutation of that array is the permutation that follows it in the sorted container. If such arrangement is not possible, the array must be rearranged as the lowest possible order (i.e., sorted in ascending order).For example, the next permutation of arr = [1,2,3] is [1,3,2].Similarly, the next permutation of arr = [2,3,1] is [3,1,2].While the next permutation of arr = [3,2,1] is [1,2,3] because [3,2,1] does not have a lexicographical larger rearrangement.Given an array of integers nums, find the next permutation of nums.The replacement must be in place and use only constant extra memory. Example 1:Input: nums = [1,2,3]Output: [1,3,2]Example 2:Input: nums = [3,2,1]Output: [1,2,3]Example 3:Input: nums = [1,1,5]Output: [1,5,1] Constraints:1 <= nums.length <= 1000 <= nums[i] <= 100

Input FormatThe first and only line of input contains an integer N.Output FormatPrint the count of permutations.Constraints1 ≤ N ≤ 100ExampleInput8Output576