1到2n这2n个数平均分成两份 abs(ai-bi)=k 求个数

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探讨了将从1到2n的数均匀分为两组，使得每组排序后，相同位置的数差的绝对值至少为k的问题。通过递归和组合方法，对比两种解决方案的正确性和效率。

输入是两个正整数n和k，把从1到2n这2n个数平均分成两份(每份n个数)，每份分别排序，排序完成后的数组假设叫a和b，要求abs(ai-bi)>=k, (相同位置的数差的绝对值不少于k),输出有多少种分法。

Backtrack is straighforward, welcome other better solutions:

from itertools import combinations

class Solution:
    def cal_ways(self, n, k):
        def dfs(toset, sta, y):
            nonlocal k,n
            l = len(sta)
            if (l == n and y == n):
                return 1
            elif (l == y and l+k <= n):
                return 2*dfs(k+toset, sta+[i+toset for i in range(k)], y)
            elif (l == y and l+k > n):
                return 0
            else:
                ret = 0
                if (l < n):
                    ret += dfs(toset+1, sta+[toset], y)
                if ((abs(sta[y]-toset) >= k)):
                    ret += dfs(toset+1, sta, y+1)
                return ret
        res = dfs(1+k, [i+1 for i in range(k)], 0)
        return res

    def cal_direct(self, n, k):
        total = [i+1 for i in range(n*2)]
        cnt,xycnt = 0,0

        for comb in combinations([i+1 for i in range(n*2)], n):
            a = list(comb)
            b = sorted(list(set(total).difference(set(a))))

            if (all(abs(a[i]-b[i]) >= k for i in range(n))):
                cnt += 1
        return cnt >> 1

s = Solution()


for i in range(1,10):
    for j in range(1,i+1):
        res1 = s.cal_ways(i, j)
        res2 = s.cal_direct(i, j)
        if (res1 != res2):
            print("n={0} k={1} res1={2} res2={3}".format(i,j,res1,res2))
print("Finished!!!")

Dependencies are the major difficulty. E.g., n=7, k=3, after arrangement of [1,2,3],the arrangement of 4,5,6 has 3 scenarios:

a=[1, 2, 3, 4, 7, 8, 9] b=[5, 6, 10, 11, 12, 13, 14]
a=[1, 2, 3, 4, 7, 8, 10] b=[5, 6, 9, 11, 12, 13, 14]
a=[1, 2, 3, 4, 7, 8, 11] b=[5, 6, 9, 10, 12, 13, 14]
a=[1, 2, 3, 4, 7, 9, 10] b=[5, 6, 8, 11, 12, 13, 14]
a=[1, 2, 3, 4, 7, 9, 11] b=[5, 6, 8, 10, 12, 13, 14]
a=[1, 2, 3, 4, 7, 10, 11] b=[5, 6, 8, 9, 12, 13, 14]
a=[1, 2, 3, 4, 8, 9, 10] b=[5, 6, 7, 11, 12, 13, 14]
a=[1, 2, 3, 4, 8, 9, 11] b=[5, 6, 7, 10, 12, 13, 14]
a=[1, 2, 3, 4, 8, 10, 11] b=[5, 6, 7, 9, 12, 13, 14]
a=[1, 2, 3, 4, 9, 10, 11] b=[5, 6, 7, 8, 12, 13, 14]
a=[1, 2, 3, 4, 12, 13, 14] b=[5, 6, 7, 8, 9, 10, 11]

a=[1, 2, 3, 5, 7, 8, 9] b=[4, 6, 10, 11, 12, 13, 14]
a=[1, 2, 3, 5, 7, 8, 10] b=[4, 6, 9, 11, 12, 13, 14]
a=[1, 2, 3, 5, 7, 8, 11] b=[4, 6, 9, 10, 12, 13, 14]
a=[1, 2, 3, 5, 7, 9, 10] b=[4, 6, 8, 11, 12, 13, 14]
a=[1, 2, 3, 5, 7, 9, 11] b=[4, 6, 8, 10, 12, 13, 14]
a=[1, 2, 3, 5, 7, 10, 11] b=[4, 6, 8, 9, 12, 13, 14]
a=[1, 2, 3, 5, 8, 9, 10] b=[4, 6, 7, 11, 12, 13, 14]
a=[1, 2, 3, 5, 8, 9, 11] b=[4, 6, 7, 10, 12, 13, 14]
a=[1, 2, 3, 5, 8, 10, 11] b=[4, 6, 7, 9, 12, 13, 14]
a=[1, 2, 3, 5, 9, 10, 11] b=[4, 6, 7, 8, 12, 13, 14]
a=[1, 2, 3, 5, 12, 13, 14] b=[4, 6, 7, 8, 9, 10, 11]

a=[1, 2, 3, 6, 7, 8, 9] b=[4, 5, 10, 11, 12, 13, 14]
a=[1, 2, 3, 6, 7, 8, 10] b=[4, 5, 9, 11, 12, 13, 14]
a=[1, 2, 3, 6, 7, 8, 11] b=[4, 5, 9, 10, 12, 13, 14]
a=[1, 2, 3, 6, 7, 9, 10] b=[4, 5, 8, 11, 12, 13, 14]
a=[1, 2, 3, 6, 7, 9, 11] b=[4, 5, 8, 10, 12, 13, 14]
a=[1, 2, 3, 6, 7, 10, 11] b=[4, 5, 8, 9, 12, 13, 14]
a=[1, 2, 3, 6, 8, 9, 10] b=[4, 5, 7, 11, 12, 13, 14]
a=[1, 2, 3, 6, 8, 9, 11] b=[4, 5, 7, 10, 12, 13, 14]
a=[1, 2, 3, 6, 8, 10, 11] b=[4, 5, 7, 9, 12, 13, 14]