2019 ICPC 上海网络赛 D. Counting Sequences I

解法：看着n的范围，队友的第一反应打表。就不能好好dfs正常输出结果吗？这不比你打表容易多了？接下来大力爆搜加剪枝，TLE。
任意一个数都秒出，你和我说TLE？那没办法了，打表安排上。

下面是AC代码：

#include 
using namespace std;
int a[3005]={0,0,1,6,12,40,30,84,224,144,135,715,627,1326,1274,420,480,2720,5202,16530,3800,2590,924,16951,552,9300,68575,82134,21168,24360,165300,3720,45632,199584,2244,22015,2520,29304,77330,57798,398320,1285760,4255062,81270,41624,48510,97290,1479889,106032,11760,180075,132600,3254004,429936,1267866,38115,2293060,20948298,40667106,299425,208860,2325320,344162,7394058,7636608,82875520,8580,305118,4013836,171258,173880,11850965,70320672,1866318,14396330,4872900,17100,895356,468468,499122,252800,20509200,7012845,1391827,578676,25229190,26418340,211474989,784847184,497121123,15339150,49140,393484,1972158,821748,2924575,53598240,55872,45638306,15086610,753390000,50040450,417498852,18788230,1649648,1168440,1202040,63736369,2461428,68719050,162308630,564673856,24864,5916680,12882,83753235,89751520,869785371,124424274,827222264,916501173,142157979,2679303,3661464,39416252,40718500,31500,4096512,135201152,132178560,5399940,191928100,764103895,303683688,755446880,403450390,878166801,7620488,63013698,181404174,188003060,1490370,40044,7198477,2965248,73807320,222581380,897305600,3241644,486312756,838638473,668099857,87240552,277345965,3652110,279481895,569777520,909958004,17438618,102598248,6080160,331917600,704201656,12886128,6522608,11108130,856576600,6916639,773139192,393768648,453484704,5029110,10088316,597113652,30102,2755725,479740800,6824611,490800468,6727676,453731610,810770040,174453052,127995139,76742980,579462920,6400260,197469008,621276832,63131059,3519180,655161515,315686842,28570176,246855106,708291090,7567560,468395047,15329358,7998606,803840600,57548093,624161197,809432154,689858065,197982428,630455180,357065367,320910720,936082576,87780,963476640,74767476,379431305,9754548,380663735,10031040,156955682,25639852,20815512,5347980,198166963,886901487,225570529,242056473,501024386,533868153,603037815,920524624,245275990,718210895,212520,12540528,543375195,461026027,32361615,540843679,392427831,168324999,208713599,700359503,28225920,380401421,67909312,81646278,630637420,1293931,889031838,949548624,259817649,549967486,898626104,686911972,56583450,24419560,817851652,16842240,891525490,17040642,862442760,43568980,561311662,17984466,437032941,809577120,18959160,388056851,262408964,560202110,38713404,657023276,821728799,818922105,190585256,635328541,948130242,157183696,704797050,309867908,277830386,51523419,134870012,749747645,184738215,448245518,318027562,275886879,333883532,395070958,48524256,475894014,49114980,236370917,784569639,979573108,921821882,773970379,110038107,105145128,235499832,26910000,64267522,208632348,110356236,214584532,461333294,560051640,903997718,590609103,218012655,780213045,692419287,439786580,279209594,383153779,93074310,814843017,988535918,205584668,145543973,54218796,420841210,489026136,101531951,476789328,74482200,664404815,556029323,35179968,106293320,217140,72747180,127035152,215692694,111253064,388964221,61191507,848268878,659514248,487518216,265388569,771127213,748161940,681164077,541338475,70311036,85182422,882943718,804827001,59357303,21376250,43611750,43737408,807625711,44111586,713948361,860057995,984432333,137008032,883198273,305220824,581376307,283279826,758860975,670467540,619538990,874650824,656833322,504482780,851303417,791600048,321615528,207289111,969710687,224897578,969569932,975009846,12704182,147213128,181988425,550724752,138047730,536076394,19423120,474768550,534347422,277774654,873715625,305094536,448946783,855709629,60538530,209522824,693786590,927479183,18760111,123571800,467930686,548061617,339961478,377197825,929122116,510292238,260505648,898560496,606820688,73823363,167968493,753754165,643514715,617757502,468134774,139190904,429596203,499872283,804316856,158784988,500436898,330820304,833779290,309908865,798032191,19144628,330236216,227597688,791835097,548987726,78035958,933622580,464458901,606535622,336637759,837971159,466100351,501035131,722499131,998404247,40098260,401772714,106748090,42495200,86735880,877149,383556205,196692,263769300,896584125,250541410,706926755,322185268,663522473,595974710,915622764,342038221,55675413,585447394,719165182,809889789,569918083,97217522,72682038,663560186,103370673,909754905,634643969,791101815,701424289,688651692,481800017,873272360,331481464,944159323,607048176,450140201,307580852,633067279,507471325,680972531,932698257,847108034,329935200,841861733,931230222,86653572,298186557,317306433,382511484,941206421,998651093,269587042,223414638,901347339,381257681,299192826,716507553,865165128,88498570,843173027,268662409,919135809,671588778,222222296,816798687,794783532,382042584,329002916,363573101,606409032,111669003,820596994,868832953,599417016,293953140,806616576,764889370,63761101,853583648,276375792,65522872,998575523,883199315,676000997,149580165,348663191,678678481,255020512,485229036,109282577,448477583,149990544,76877264,304513527,150568236,655124776,151988148,534097455,270093859,221569978,619992276,544538574,150812922,389192157,202124556,637466796,904314865,928886810,154958903,986123802,493231088,174281319,670239135,62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int main()
{
    int T,n;scanf("%d",&T);
    while(T--){scanf("%d",&n);printf("%d\n",a[n]);}
    return 0;
}

值得一提的是，我们打表用的是绝对正确的暴力枚举，推导出来发现ai最多有13个大于1，枚举13个数从1到n，枚举过程适当剪枝，大概在30s左右就可以打出来。

update :鉴于有人问具体是怎么打表的，我就去特地问了一下队友，毕竟后面打表的代码是他写的。
方法 ：对于任意一个n，如果小于14，直接对每一个ai进行枚举，否则，如果n大于或等于14，只枚举14位，剩下的肯定全1，因为2^14明显比3000大很多,14个2，其他全是1，乘积都大上天了,再多应该不可能。接下来进入dfs，第一点，我们默认ai是一个不下降的序列，也就是说a1≤a2≤a3…≤ai,每次成功的枚举之后，要算出它的全排列数，这个后面讲怎么算。第二点，在任意的一个时候，如果我们发现乘积比和大，就不用再枚举了，因为我们是从a1，也就是最小的数开始枚举，和不可能追上乘积。第三点，ai的枚举上届不可能超过n，比n大的话随便乘一个2，和就追不上了，同时所有ai的和不可能大于6000，等于的情况是2998个1，1个2,1个3000，也就是说枚举新的ai的时候下届为a(i-1)，上届为min(6000/乘积 , n)。因为是打表，我们就把范围都开大了很多，怎么安全怎么来。最后说一下发现 和 等于 乘积了之后算全排列，查一下我们枚举的过程中出现了哪些数字，和他们分别出现了几次。举个例子，n=3000,枚举完14个数以后发现一组解有5个1,7个2,2个3（随便举的），如果3000个数全部不一样，那么全排列就是3000！，但是现在有数一样，就要去掉重复的，1的数量要算一下是3000-14+5=2991个，那么对于这组数据全排列数就是(3000！)/(2991! * 7! * 2!)，事先把3000以内所有数的阶乘和逆元求好即可。

下面是打表代码

 #include 
 using namespace std;
 using namespace chrono;
 const int M = 1000000007;
 const int INF = 0x3f3f3f3f;
 const double PI = acos(-1);
 const double eps = 1e-8;
 #define ms(x, y) memset((x), (y), sizeof(x))
 #define mc(x, y) memcpy((x), (y), sizeof(y))
 typedef long long ll;
 typedef unsigned long long ull;
 #define fi first
 #define se second
 #define mp make_pair
 typedef pair<int, int> pii;
 typedef pair<ll, int> pli;
 #define bg begin
 #define ed end
 #define pb push_back
 #define al(x) (x).bg(), (x).ed()
 #define st(x) sort(al(x))
 #define un(x) (x).erase(unique(al(x)), (x).ed())
 #define fd(x, y) (lower_bound(al(x), (y)) - (x).bg() + 1)
 #define ls(x) ((x) << 1)
 #define rs(x) (ls(x) | 1)
 template <class T>
 bool read(T & x) {
     char c;
     while (!isdigit(c = getchar()) && c != '-' && c != EOF);
     if (c == EOF) return false;
     T flag = 1;
     if (c == '-') { flag = -1; x = 0; } else x = c - '0';
     while (isdigit(c = getchar())) x = x * 10 + c - '0';
     x *= flag;
     return true;
 }
 template <class T, class ...R>
 bool read(T & a, R & ...b) {
     if (!read(a)) return false;
     return read(b...);
 }
 mt19937 gen(steady_clock::now().time_since_epoch().count());
 ll fac[3005], ifac[3005], cnt[6005], ans;
 ll qpow(ll a, ll n, ll p) {
     ll r = 1;
     for (a %= p; n; n >>= 1, (a *= a) %= p)
         if (n & 1) (r *= a) %= p;
     return r;
 }
 const int lim = 14;
 int N;
 void dfs(int d, int p, int n, ll sum, ll prod) {
     if (d == n) {
         int t = N > lim ? N - lim : 0;
         if (sum + t == prod) {
             ll x = fac[N];
             (x *= ifac[cnt[1] + t]) %= M;
             // cout << "scheme: " << endl;
             // cout << "1: " << cnt[1] + (N > lim ? N - lim : 0) << endl;
             for (int i = 2; i < 6005; i++) {
                 (x *= ifac[cnt[i]]) %= M;
                 // if (cnt[i]) {
                 //     cout << i << ": " << cnt[i] << endl;
                 // }
             }
             (ans += x) %= M;
         }
         return;
     }
     if (prod > 6000) return;
     for (int i = p; i <= 6000 / prod; i++) {
         cnt[i]++;
         dfs(d + 1, i, n, sum + i, prod * i);
         cnt[i]--;
     }
 }
 ll C(ll n, ll m, ll p) {
     return fac[n] * ifac[m] % p * ifac[n - m] % p;
 }
 int main()
 {
     time_point<steady_clock> start = steady_clock::now();

     freopen("in.txt", "r", stdin);
     freopen("out4.txt", "w", stdout);

     fac[0] = 1;
     ifac[0] = 1;
     for (int i = 1; i < 3005; i++) {
         fac[i] = fac[i - 1] * i % M;
         ifac[i] = qpow(fac[i], M - 2, M);
     }

     int T;
     read(T);
     printf("{");
     for (N = 2; N <= 3000; N++) {
         ans = 0;

         int t = min(N, lim);
         dfs(0, 1, t, 0, 1);
        
         printf("%lld,", ans);
     }
     printf("};\n");

     cerr << endl << "------------------------------" << endl << "Time: "
          << duration<double, milli>(steady_clock::now() - start).count()
         << " ms." << endl;

    return 0;
 }