AtCoder ABC152

C - Low Elements
从前往后维护一个最长下降子序列

D - Handstand 2
设f[a][b]代表当前第一个数字为a第二个数字为b的数总个数
递推一下就可以。注意a==b的情况。

# -*- coding: utf-8 -*-
# @time     : 2023/6/2 13:30
# @file     : atcoder.py
# @software : PyCharm

import bisect
import copy
import sys
from itertools import permutations
from sortedcontainers import SortedList
from collections import defaultdict, Counter, deque
from functools import lru_cache, cmp_to_key
import heapq
import math
sys.setrecursionlimit(1000)


def main():
    items = sys.version.split()
    if items[0] == '3.10.6':
        fp = open("in.txt")
    else:
        fp = sys.stdin

    n = int(fp.readline())
    d = [[0] * 10 for _ in range(10)]
    ans = 0
    for i in range(1, n + 1):
        s = str(i)
        a, b = int(s[0]), int(s[-1])
        d[a][b] += 1
        if a != b:
            ans += d[b][a] * 2
        else:
            ans += (d[a][a] - 1) * 2 + 1
    print(ans)


if __name__ == "__main__":
    main()

E - Flatten
数论题，需要分解质因数和逆元，然而python改变了一切

# -*- coding: utf-8 -*-
# @time     : 2023/6/2 13:30
# @file     : atcoder.py
# @software : PyCharm

import bisect
import copy
import sys
from itertools import permutations
from sortedcontainers import SortedList
from collections import defaultdict, Counter, deque
from functools import lru_cache, cmp_to_key
import heapq
import math
sys.setrecursionlimit(1000)


def main():
    items = sys.version.split()
    if items[0] == '3.10.6':
        fp = open("in.txt")
    else:
        fp = sys.stdin
    n = int(fp.readline())
    a = list(map(int, fp.readline().split()))

    def lcm(x, y):
        x0, y0 = x, y
        if x0 > y0:
            x0, y0 = y0, x0
        while x0:
            x0, y0 = y0 % x0, x0
        return x * y // y0

    l = a[0]
    for i in range(1, n):
        l = lcm(l, a[i])

    ans = 0
    mod = 10 ** 9 + 7
    for i in range(n):
        ans += l // a[i]
        ans %= mod
    print(ans % mod)


if __name__ == "__main__":
    main()

F - Tree and Constraints
反过来考虑。比如下图中有两个constraint，原题求满足所有的限制的方案，需要枚举每种黑色放置的方案，很不好做。如果反过来求不满足限制的方案，即在某些限制路径上全填白色，求总的方案数，那么就比较好做。
由于不同的限制路径下边的集合会有交集，因此需要使用容斥原理计算。设取不同的限制路径的状态集合为$s$，计算状态下边的并集中边数量为$x$，不满足限制的方案数等于$$ans=\sum _{s\in S}(-1{)}^{|s|+1}{2}^{n-1-x}$$
如下图中按顺序对边和点0-index编号，
第一个状态集合是0b01，代表限制状态0
f(1)=2，因为边2可以取两种颜色，边0边1全取白
第二个状态集合是0b10，代表限制状态1
f(2)=2，同理
第三个状态集合0b11，代表两个限制状态都要取
在这种情况下，不满足条件的方案只有1个，就是三条边全取白
所以最终不满足任一约束的方案数是2+2-1=3个，因为三条边全取白的方案在0b01和0b10中都计数了一次，因此按照容斥原理予以减去

# -*- coding: utf-8 -*-
# @time     : 2023/6/2 13:30
# @file     : atcoder.py
# @software : PyCharm

import bisect
import copy
import sys
from itertools import permutations
from sortedcontainers import SortedList
from collections import defaultdict, Counter, deque
from functools import lru_cache, cmp_to_key
import heapq
import math
sys.setrecursionlimit(1000)


def main():
    items = sys.version.split()
    if items[0] == '3.10.6':
        fp = open("in.txt")
    else:
        fp = sys.stdin
    n = int(fp.readline())
    g = [[] for _ in range(n)]
    fa = [-1] * n
    dep = [0] * n
    edge_mask = {}
    for i in range(n - 1):
        u, v = map(int, fp.readline().split())
        u -= 1
        v -= 1
        g[u].append(v)
        g[v].append(u)
        edge_mask[(u, v)] = edge_mask[(v, u)] = i

    def dfs0(u, f):
        for v in g[u]:
            if f == v:
                continue
            fa[v] = u
            dep[v] = dep[u] + 1
            dfs0(v, u)

    def lca(u, v):
        while u != v:
            if dep[u] > dep[v]:
                u = fa[u]
            else:
                v = fa[v]
        return u

    dfs0(0, -1)
    m = int(fp.readline())
    con = [0] * m
    for i in range(m):
        u, v = map(int, fp.readline().split())
        u -= 1
        v -= 1
        l = lca(u, v)
        while u != l:
            fu = fa[u]
            j = edge_mask[(fu, u)]
            con[i] |= 1 << j
            u = fu
        while v != l:
            fv = fa[v]
            j = edge_mask[(fv, v)]
            con[i] |= 1 << j
            v = fv

    bit_count = [0] * (1 << m)
    bit = [0] * (1 << m)
    for i in range(m):
        bit[1 << i] = i
    for i in range(1, 1 << m):
        if i & 1 == 1:
            bit_count[i] = bit_count[i >> 1] + 1
        else:
            bit_count[i] = bit_count[i >> 1]

    def get_bit_count(val):
        ret = 0
        while val:
            ret += val & 1
            val >>= 1
        return ret

    union = [0] * (1 << m)
    union[0] = 0
    temp = 0
    for i in range(1, 1 << m):
        lb = i & -i
        j = bit[lb]
        union[i] = union[i - lb] | con[j]
        x = get_bit_count(union[i])
        t = 1 << (n - 1 - x)
        if bit_count[i] & 1:
            temp += t
        else:
            temp -= t
    ans = (1 << n - 1) - temp
    print(ans)


if __name__ == "__main__":
    main()