use std::io::Write; use std::collections::*; type Map = BTreeMap; type Set = BTreeSet; type Deque = VecDeque; fn main() { input!(n: usize, m: usize); type M = ModInt<998244353>; let pc = Precalc::new(n); let mut f = vec![M::zero(); n + 1]; for i in 0..=n { f[i] = M::from(m.saturating_sub(i)).pow(n as u64) * pc.ifact(i); } let mut g = vec![M::zero(); n + 1]; for i in 0..=n { g[i] = pc.ifact(i); if i & 1 == 1 { g[i] = -g[i]; } } let h = f.convolution(&g); let mut ans = M::zero(); for i in 1..=n { if i & 1 == 1 { ans -= M::from(i).pow(2) * h[i] * pc.fact(i); } else { ans += M::from(i).pow(2) * h[i] * pc.fact(i); } } println!("{}", ans); } // ---------- begin input macro ---------- // reference: https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 #[macro_export] macro_rules! input { (source = $s:expr, $($r:tt)*) => { let mut iter = $s.split_whitespace(); input_inner!{iter, $($r)*} }; ($($r:tt)*) => { let s = { use std::io::Read; let mut s = String::new(); std::io::stdin().read_to_string(&mut s).unwrap(); s }; let mut iter = s.split_whitespace(); input_inner!{iter, $($r)*} }; } #[macro_export] macro_rules! input_inner { ($iter:expr) => {}; ($iter:expr, ) => {}; ($iter:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($iter, $t); input_inner!{$iter $($r)*} }; } #[macro_export] macro_rules! read_value { ($iter:expr, ( $($t:tt),* )) => { ( $(read_value!($iter, $t)),* ) }; ($iter:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($iter, $t)).collect::>() }; ($iter:expr, chars) => { read_value!($iter, String).chars().collect::>() }; ($iter:expr, bytes) => { read_value!($iter, String).bytes().collect::>() }; ($iter:expr, usize1) => { read_value!($iter, usize) - 1 }; ($iter:expr, $t:ty) => { $iter.next().unwrap().parse::<$t>().expect("Parse error") }; } // ---------- end input macro ---------- use std::ops::*; // ---------- begin trait ---------- pub trait Zero: Sized + Add { fn zero() -> Self; fn is_zero(&self) -> bool; } pub trait One: Sized + Mul { fn one() -> Self; fn is_one(&self) -> bool; } pub trait SemiRing: Zero + One {} pub trait Ring: SemiRing + Sub + Neg {} pub trait Field: Ring + Div {} impl SemiRing for T where T: Zero + One {} impl Ring for T where T: SemiRing + Sub + Neg {} impl Field for T where T: Ring + Div {} // ---------- end trait ---------- // ---------- begin modint ---------- pub const fn pow_mod(mut r: u32, mut n: u32, m: u32) -> u32 { let mut t = 1; while n > 0 { if n & 1 == 1 { t = (t as u64 * r as u64 % m as u64) as u32; } r = (r as u64 * r as u64 % m as u64) as u32; n >>= 1; } t } pub const fn primitive_root(p: u32) -> u32 { let mut m = p - 1; let mut f = [1; 30]; let mut k = 0; let mut d = 2; while d * d <= m { if m % d == 0 { f[k] = d; k += 1; } while m % d == 0 { m /= d; } d += 1; } if m > 1 { f[k] = m; k += 1; } let mut g = 1; while g < p { let mut ok = true; let mut i = 0; while i < k { ok &= pow_mod(g, (p - 1) / f[i], p) > 1; i += 1; } if ok { break; } g += 1; } g } pub const fn is_prime(n: u32) -> bool { if n <= 1 { return false; } let mut d = 2; while d * d <= n { if n % d == 0 { return false; } d += 1; } true } #[derive(Clone, Copy, PartialEq, Eq)] pub struct ModInt(u32); impl ModInt<{ M }> { const REM: u32 = { let mut t = 1u32; let mut s = !M + 1; let mut n = !0u32 >> 2; while n > 0 { if n & 1 == 1 { t = t.wrapping_mul(s); } s = s.wrapping_mul(s); n >>= 1; } t }; const INI: u64 = ((1u128 << 64) % M as u128) as u64; const IS_PRIME: () = assert!(is_prime(M)); const PRIMITIVE_ROOT: u32 = primitive_root(M); const ORDER: usize = 1 << (M - 1).trailing_zeros(); const fn reduce(x: u64) -> u32 { let _ = Self::IS_PRIME; let b = (x as u32 * Self::REM) as u64; let t = x + b * M as u64; let mut c = (t >> 32) as u32; if c >= M { c -= M; } c as u32 } const fn multiply(a: u32, b: u32) -> u32 { Self::reduce(a as u64 * b as u64) } pub const fn new(v: u32) -> Self { assert!(v < M); Self(Self::reduce(v as u64 * Self::INI)) } pub const fn const_mul(&self, rhs: Self) -> Self { Self(Self::multiply(self.0, rhs.0)) } pub const fn pow(&self, mut n: u64) -> Self { let mut t = Self::new(1); let mut r = *self; while n > 0 { if n & 1 == 1 { t = t.const_mul(r); } r = r.const_mul(r); n >>= 1; } t } pub const fn inv(&self) -> Self { assert!(self.0 != 0); self.pow(M as u64 - 2) } pub const fn get(&self) -> u32 { Self::reduce(self.0 as u64) } pub const fn zero() -> Self { Self::new(0) } pub const fn one() -> Self { Self::new(1) } } impl Add for ModInt<{ M }> { type Output = Self; fn add(self, rhs: Self) -> Self::Output { let mut v = self.0 + rhs.0; if v >= M { v -= M; } Self(v) } } impl Sub for ModInt<{ M }> { type Output = Self; fn sub(self, rhs: Self) -> Self::Output { let mut v = self.0 - rhs.0; if self.0 < rhs.0 { v += M; } Self(v) } } impl Mul for ModInt<{ M }> { type Output = Self; fn mul(self, rhs: Self) -> Self::Output { self.const_mul(rhs) } } impl Div for ModInt<{ M }> { type Output = Self; fn div(self, rhs: Self) -> Self::Output { self * rhs.inv() } } impl AddAssign for ModInt<{ M }> { fn add_assign(&mut self, rhs: Self) { *self = *self + rhs; } } impl SubAssign for ModInt<{ M }> { fn sub_assign(&mut self, rhs: Self) { *self = *self - rhs; } } impl MulAssign for ModInt<{ M }> { fn mul_assign(&mut self, rhs: Self) { *self = *self * rhs; } } impl DivAssign for ModInt<{ M }> { fn div_assign(&mut self, rhs: Self) { *self = *self / rhs; } } impl Neg for ModInt<{ M }> { type Output = Self; fn neg(self) -> Self::Output { if self.0 == 0 { self } else { Self(M - self.0) } } } impl std::fmt::Display for ModInt<{ M }> { fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result { write!(f, "{}", self.get()) } } impl std::fmt::Debug for ModInt<{ M }> { fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result { write!(f, "{}", self.get()) } } impl std::str::FromStr for ModInt<{ M }> { type Err = std::num::ParseIntError; fn from_str(s: &str) -> Result { let val = s.parse::()?; Ok(ModInt::new(val)) } } impl From for ModInt<{ M }> { fn from(val: usize) -> ModInt<{ M }> { ModInt::new((val % M as usize) as u32) } } // ---------- end modint ---------- // ---------- begin precalc ---------- pub struct Precalc { fact: Vec>, ifact: Vec>, inv: Vec>, } impl Precalc { pub fn new(size: usize) -> Self { let mut fact = vec![ModInt::one(); size + 1]; let mut ifact = vec![ModInt::one(); size + 1]; let mut inv = vec![ModInt::one(); size + 1]; for i in 2..=size { fact[i] = fact[i - 1] * ModInt::from(i); } ifact[size] = fact[size].inv(); for i in (2..=size).rev() { inv[i] = ifact[i] * fact[i - 1]; ifact[i - 1] = ifact[i] * ModInt::from(i); } Self { fact, ifact, inv } } pub fn fact(&self, n: usize) -> ModInt { self.fact[n] } pub fn ifact(&self, n: usize) -> ModInt { self.ifact[n] } pub fn inv(&self, n: usize) -> ModInt { assert!(0 < n); self.inv[n] } pub fn perm(&self, n: usize, k: usize) -> ModInt { if k > n { return ModInt::zero(); } self.fact[n] * self.ifact[n - k] } pub fn binom(&self, n: usize, k: usize) -> ModInt { if n < k { return ModInt::zero(); } self.fact[n] * self.ifact[k] * self.ifact[n - k] } } // ---------- end precalc ---------- impl Zero for ModInt<{ M }> { fn zero() -> Self { Self::zero() } fn is_zero(&self) -> bool { self.0 == 0 } } impl One for ModInt<{ M }> { fn one() -> Self { Self::one() } fn is_one(&self) -> bool { self.get() == 1 } } // ---------- begin array op ---------- struct NTTPrecalc { sum_e: [ModInt<{ M }>; 30], sum_ie: [ModInt<{ M }>; 30], } impl NTTPrecalc<{ M }> { const fn new() -> Self { let cnt2 = (M - 1).trailing_zeros() as usize; let root = ModInt::new(ModInt::<{ M }>::PRIMITIVE_ROOT); let zeta = root.pow((M - 1) as u64 >> cnt2); let mut es = [ModInt::zero(); 30]; let mut ies = [ModInt::zero(); 30]; let mut sum_e = [ModInt::zero(); 30]; let mut sum_ie = [ModInt::zero(); 30]; let mut e = zeta; let mut ie = e.inv(); let mut i = cnt2; while i >= 2 { es[i - 2] = e; ies[i - 2] = ie; e = e.const_mul(e); ie = ie.const_mul(ie); i -= 1; } let mut now = ModInt::one(); let mut inow = ModInt::one(); let mut i = 0; while i < cnt2 - 1 { sum_e[i] = es[i].const_mul(now); sum_ie[i] = ies[i].const_mul(inow); now = ies[i].const_mul(now); inow = es[i].const_mul(inow); i += 1; } Self { sum_e, sum_ie } } } struct NTTPrecalcHelper; impl NTTPrecalcHelper { const A: NTTPrecalc = NTTPrecalc::new(); } pub trait ArrayAdd { type Item; fn add(&self, rhs: &[Self::Item]) -> Vec; } impl ArrayAdd for [T] where T: Zero + Copy, { type Item = T; fn add(&self, rhs: &[Self::Item]) -> Vec { let mut c = vec![T::zero(); self.len().max(rhs.len())]; c[..self.len()].copy_from_slice(self); c.add_assign(rhs); c } } pub trait ArrayAddAssign { type Item; fn add_assign(&mut self, rhs: &[Self::Item]); } impl ArrayAddAssign for [T] where T: Add + Copy, { type Item = T; fn add_assign(&mut self, rhs: &[Self::Item]) { assert!(self.len() >= rhs.len()); self.iter_mut().zip(rhs).for_each(|(x, a)| *x = *x + *a); } } impl ArrayAddAssign for Vec where T: Zero + Add + Copy, { type Item = T; fn add_assign(&mut self, rhs: &[Self::Item]) { if self.len() < rhs.len() { self.resize(rhs.len(), T::zero()); } self.as_mut_slice().add_assign(rhs); } } pub trait ArraySub { type Item; fn sub(&self, rhs: &[Self::Item]) -> Vec; } impl ArraySub for [T] where T: Zero + Sub + Copy, { type Item = T; fn sub(&self, rhs: &[Self::Item]) -> Vec { let mut c = vec![T::zero(); self.len().max(rhs.len())]; c[..self.len()].copy_from_slice(self); c.sub_assign(rhs); c } } pub trait ArraySubAssign { type Item; fn sub_assign(&mut self, rhs: &[Self::Item]); } impl ArraySubAssign for [T] where T: Sub + Copy, { type Item = T; fn sub_assign(&mut self, rhs: &[Self::Item]) { assert!(self.len() >= rhs.len()); self.iter_mut().zip(rhs).for_each(|(x, a)| *x = *x - *a); } } impl ArraySubAssign for Vec where T: Zero + Sub + Copy, { type Item = T; fn sub_assign(&mut self, rhs: &[Self::Item]) { if self.len() < rhs.len() { self.resize(rhs.len(), T::zero()); } self.as_mut_slice().sub_assign(rhs); } } pub trait ArrayDot { type Item; fn dot(&self, rhs: &[Self::Item]) -> Vec; } impl ArrayDot for [T] where T: Mul + Copy, { type Item = T; fn dot(&self, rhs: &[Self::Item]) -> Vec { assert!(self.len() == rhs.len()); self.iter().zip(rhs).map(|p| *p.0 * *p.1).collect() } } pub trait ArrayDotAssign { type Item; fn dot_assign(&mut self, rhs: &[Self::Item]); } impl ArrayDotAssign for [T] where T: MulAssign + Copy, { type Item = T; fn dot_assign(&mut self, rhs: &[Self::Item]) { assert!(self.len() == rhs.len()); self.iter_mut().zip(rhs).for_each(|(x, a)| *x *= *a); } } pub trait ArrayMul { type Item; fn mul(&self, rhs: &[Self::Item]) -> Vec; } impl ArrayMul for [T] where T: Zero + One + Copy, { type Item = T; fn mul(&self, rhs: &[Self::Item]) -> Vec { if self.is_empty() || rhs.is_empty() { return vec![]; } let mut res = vec![T::zero(); self.len() + rhs.len() - 1]; for (i, a) in self.iter().enumerate() { for (res, b) in res[i..].iter_mut().zip(rhs.iter()) { *res = *res + *a * *b; } } res } } // transform でlen=1を指定すればNTTになる pub trait ArrayConvolution { type Item; fn transform(&mut self, len: usize); fn inverse_transform(&mut self, len: usize); fn convolution(&self, rhs: &[Self::Item]) -> Vec; } impl ArrayConvolution for [ModInt<{ M }>] { type Item = ModInt<{ M }>; fn transform(&mut self, len: usize) { let f = self; let n = f.len(); let k = (n / len).trailing_zeros() as usize; assert!(len << k == n); assert!(k <= ModInt::<{ M }>::ORDER); let pre = &NTTPrecalcHelper::<{ M }>::A; for ph in 1..=k { let p = len << (k - ph); let mut now = ModInt::one(); for (i, f) in f.chunks_exact_mut(2 * p).enumerate() { let (x, y) = f.split_at_mut(p); for (x, y) in x.iter_mut().zip(y.iter_mut()) { let l = *x; let r = *y * now; *x = l + r; *y = l - r; } now *= pre.sum_e[(!i).trailing_zeros() as usize]; } } } fn inverse_transform(&mut self, len: usize) { let f = self; let n = f.len(); let k = (n / len).trailing_zeros() as usize; assert!(len << k == n); assert!(k <= ModInt::<{ M }>::ORDER); let pre = &NTTPrecalcHelper::<{ M }>::A; for ph in (1..=k).rev() { let p = len << (k - ph); let mut inow = ModInt::one(); for (i, f) in f.chunks_exact_mut(2 * p).enumerate() { let (x, y) = f.split_at_mut(p); for (x, y) in x.iter_mut().zip(y.iter_mut()) { let l = *x; let r = *y; *x = l + r; *y = (l - r) * inow; } inow *= pre.sum_ie[(!i).trailing_zeros() as usize]; } } let ik = ModInt::new(2).inv().pow(k as u64); for f in f.iter_mut() { *f *= ik; } } fn convolution(&self, rhs: &[Self::Item]) -> Vec { if self.len().min(rhs.len()) <= 32 { return self.mul(rhs); } const PARAM: usize = 10; let size = self.len() + rhs.len() - 1; let mut k = 0; while (size + (1 << k) - 1) >> k > PARAM { k += 1; } let len = (size + (1 << k) - 1) >> k; let mut f = vec![ModInt::zero(); len << k]; let mut g = vec![ModInt::zero(); len << k]; f[..self.len()].copy_from_slice(self); g[..rhs.len()].copy_from_slice(rhs); f.transform(len); g.transform(len); let mut buf = [ModInt::zero(); 2 * PARAM - 1]; let buf = &mut buf[..(2 * len - 1)]; let pre = &NTTPrecalcHelper::<{ M }>::A; let mut now = ModInt::one(); for (i, (f, g)) in f .chunks_exact_mut(2 * len) .zip(g.chunks_exact(2 * len)) .enumerate() { let mut r = now; for (f, g) in f.chunks_exact_mut(len).zip(g.chunks_exact(len)) { buf.fill(ModInt::zero()); for (i, f) in f.iter().enumerate() { for (buf, g) in buf[i..].iter_mut().zip(g.iter()) { *buf = *buf + *f * *g; } } f.copy_from_slice(&buf[..len]); for (f, buf) in f.iter_mut().zip(buf[len..].iter()) { *f = *f + r * *buf; } r = -r; } now *= pre.sum_e[(!i).trailing_zeros() as usize]; } f.inverse_transform(len); f.truncate(self.len() + rhs.len() - 1); f } } // ---------- end array op ----------