Title
Parsing
Consider how to judge whether a sequence satisfies the condition. Let $ c_i $ represent the number of occurrences of the number $ i $, then: $$ \ sum_ {i = 1} ^ D \ left \ lfloor \ frac {c_i} {2} \ right \ rfloor \ geqslant m \\ \ sum_ {i = 1} ^ D \ frac {c_i- (c_i \% 2)} {2} \ geqslant m \\ n- \ sum_ {i = 1} ^ D (c_i \% 2) \ geqslant 2m \\ \ sum_ {i = 1} ^ D (c_i \% 2) \ leqslant n-2m $$
When $ n-2m <0 $, output $ 0 $; when $ n-2m \ geqslant D $, output $ D ^ n $
Let $ g_i $ mean that exactly the number of occurrences of $ i $ is odd, and $ f_i $ means that at least the occurrences of $ i $ are odd, and $ f_i = \ sum_ {j = i} \ binom {j} { i} g_j $
If we find $ f $ then we can find $ g $ by binomial inversion, so now we need to find a way to find $ f $
Consider generating functions, let $ A (x) = \ sum_ {i = 0} ^ {\ infty} [i \% 2 == 1] x ^ i = \ frac {e ^ xe ^ {-x}} {2 } $, $ B (x) = \ sum_ {i = 0} ^ {\ infty} x ^ i = e ^ x $
钦定有$i$个数出现奇数次，剩下的数出现任意次数，则有：$$\begin{align*}f_i&=\binom{D}{i}n![x^n]A^iB^{D-i}\\&=\binom{D}{i}n^ie^{(D-i)x}\\&=\frac{D!}{i!(D-i)!}n!*\frac{1}{2^i}[x^n]\sum_{j=0}^i\binom{i}{j}(-1)^{i-j}e^{jx}e^{-(i-j)x}e^{(D-i)x}\\&=\frac{D!n!}{i!(D-i)!2^i}*[x^n]\sum_{j=0}^i\binom{i}{j}(-1)^{i-j}e^{(D-2i+2j)x}\\&=\frac{D!n!}{i!(D-i)!2^i}\sum_{j=0}^i\frac{i!}{j!(i-j)!}*(-1)^{i-j}*\frac{(D-2i+2j)^n}{n!}\\&=\frac{D!}{(D-i)!2^i}\sum_{j=0}^i\frac{(-1)^{i-j}(D-2i+2j)^n}{(i-j)!}*\frac{1}{j!}\end{align*}$$
Convolve to find $ f $
然后二项式反演：$$\begin{align*}g_i&=\sum_{j=i}^D(-1)^{j-i}\binom{j}{i}f_j\\&=\frac{1}{i!}\sum_{j=i}^D\frac{(-1)^{j-i}}{(j-i)!}*j!*f_j\end{align*}$$
The final answer is: $ Ans = \ sum_ {i = 0} ^ {n-2m} g_i $$
$O(D \log D)$
Code:
#include<cstdio> #include<iostream> #include<algorithm> #include<cstring> using namespace std; typedef long long ll; const int maxn = 200005, mod = 998244353, g = 3; int add(int x, int y) { return x + y < mod? x + y: x + y - mod; } int rdc(int x, int y) { return x - y < 0? x - y + mod: x - y; } ll qpow(ll x, int y) { ll ret = 1; while(y) { if(y&1) ret = ret * x % mod; x = x * x % mod; y >>= 1; } return ret; } int D, n, m; int lim, bit, rev[maxn<<1]; ll fac[maxn], fnv[maxn]; ll ginv, f[maxn<<1], A[maxn<<1], B[maxn<<1]; void init() { ginv = qpow(g, mod - 2); fac[0] = 1; for(int i = 1; i <= D; ++i) fac[i] = fac[i-1] * i % mod; fnv[D] = qpow(fac[D], mod - 2); for(int i = D - 1; i >= 0; --i) fnv[i] = fnv[i+1] * (i + 1) % mod; } void NTT_init(int x) { lim = 1; bit = 0; while(lim <= x) { lim <<= 1; ++ bit; } for(int i = 1; i < lim; ++i) rev[i] = (rev[i>>1] >> 1) | ((i & 1) << (bit - 1)); } void NTT(ll *x, int y) { for(int i = 1; i < lim; ++i) if(i < rev[i]) swap(x[i], x[rev[i]]); ll wn, w, u, v; for(int i = 1; i < lim; i <<= 1) { wn = qpow((y == 1)? g: ginv, (mod - 1) / (i << 1)); for(int j = 0; j < lim; j += (i << 1)) { w = 1; for(int k = 0; k < i; ++k) { u = x[j+k]; v = x[j+k+i] * w % mod; x[j+k] = add(u, v); x[j+k+i] = rdc(u, v); w = w * wn % mod; } } } if(y == -1) { ll linv = qpow(lim, mod - 2); for(int i = 0; i < lim; ++i) x[i] = x[i] * linv % mod; } } int main() { scanf("%d%d%d", &D, &n, &m); if(n - 2 * m < 0) { printf("0"); return 0; } if(n - 2 * m >= D) { printf("%lld", qpow(D, n)); return 0; } init(); for(int i = 0; i <= D; ++i) { A[i] = qpow(rdc(D, 2 * i), n) * fnv[i] % mod; A[i] = ((i & 1)? rdc(0, A[i]): A[i]); B[i] = fnv[i]; } NTT_init(D << 1); NTT(A, 1); NTT(B, 1); for(int i = 0; i < lim; ++i) A[i] = A[i] * B[i] % mod; NTT(A, -1); for(int i = 0; i <= D; ++i) A[i] = ((A[i] * fac[D] % mod) * qpow(qpow(2, i), mod - 2) % mod) * fnv[D-i] % mod; for(int i = D + 1; i < lim; ++i) A[i] = 0; for(int i = 0; i <= D; ++i) { A[i] = A[i] * fac[i] % mod; f[D-i] = ((i & 1)? rdc(0, fnv[i]): fnv[i]); } NTT(A, 1); NTT(f, 1); for(int i = 0; i < lim; ++i) f[i] = f[i] * A[i] % mod; NTT(f, -1); int ans = 0; for(int i = 0; i <= n - 2 * m; ++i) ans = add(ans, f[D+i] * fnv[i] % mod); printf("%d", ans); return 0; }