use std::collections::*; // https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 macro_rules! input { ($($r:tt)*) => { let stdin = std::io::stdin(); let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock())); let mut next = move || -> String{ bytes.by_ref().map(|r|r.unwrap() as char) .skip_while(|c|c.is_whitespace()) .take_while(|c|!c.is_whitespace()) .collect() }; input_inner!{next, $($r)*} }; } macro_rules! input_inner { ($next:expr) => {}; ($next:expr,) => {}; ($next:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($next, $t); input_inner!{$next $($r)*} }; } macro_rules! read_value { ($next:expr, chars) => { read_value!($next, String).chars().collect::>() }; ($next:expr, $t:ty) => ($next().parse::<$t>().expect("Parse error")); } /// Verified by https://atcoder.jp/contests/abc198/submissions/21774342 mod mod_int { use std::ops::*; pub trait Mod: Copy { fn m() -> i64; } #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)] pub struct ModInt { pub x: i64, phantom: ::std::marker::PhantomData } impl ModInt { // x >= 0 pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) } fn new_internal(x: i64) -> Self { ModInt { x: x, phantom: ::std::marker::PhantomData } } pub fn pow(self, mut e: i64) -> Self { debug_assert!(e >= 0); let mut sum = ModInt::new_internal(1); let mut cur = self; while e > 0 { if e % 2 != 0 { sum *= cur; } cur *= cur; e /= 2; } sum } #[allow(dead_code)] pub fn inv(self) -> Self { self.pow(M::m() - 2) } } impl Default for ModInt { fn default() -> Self { Self::new_internal(0) } } impl>> Add for ModInt { type Output = Self; fn add(self, other: T) -> Self { let other = other.into(); let mut sum = self.x + other.x; if sum >= M::m() { sum -= M::m(); } ModInt::new_internal(sum) } } impl>> Sub for ModInt { type Output = Self; fn sub(self, other: T) -> Self { let other = other.into(); let mut sum = self.x - other.x; if sum < 0 { sum += M::m(); } ModInt::new_internal(sum) } } impl>> Mul for ModInt { type Output = Self; fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) } } impl>> AddAssign for ModInt { fn add_assign(&mut self, other: T) { *self = *self + other; } } impl>> SubAssign for ModInt { fn sub_assign(&mut self, other: T) { *self = *self - other; } } impl>> MulAssign for ModInt { fn mul_assign(&mut self, other: T) { *self = *self * other; } } impl Neg for ModInt { type Output = Self; fn neg(self) -> Self { ModInt::new(0) - self } } impl ::std::fmt::Display for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { self.x.fmt(f) } } impl From for ModInt { fn from(x: i64) -> Self { Self::new(x) } } } // mod mod_int macro_rules! define_mod { ($struct_name: ident, $modulo: expr) => { #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)] struct $struct_name {} impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } } } } const MOD: i64 = 998_244_353; define_mod!(P, MOD); type MInt = mod_int::ModInt

; // Depends on MInt.rs fn fact_init(w: usize) -> (Vec, Vec) { let mut fac = vec![MInt::new(1); w]; let mut invfac = vec![0.into(); w]; for i in 1..w { fac[i] = fac[i - 1] * i as i64; } invfac[w - 1] = fac[w - 1].inv(); for i in (0..w - 1).rev() { invfac[i] = invfac[i + 1] * (i as i64 + 1); } (fac, invfac) } // https://yukicoder.me/problems/no/2388 (3) // u を固定し、u + (長さ m - |u| 以下のの任意の文字列) の中で条件を満たすものを数える問題を 26N 回程度解く問題に帰着できる。 // これの答えを f(u) と置く。f(u) の中で全体が v 種類になるものの個数を f_v(u) と置く。 // u に含まれる文字の種類を x 種類とすると f_v(u) = \sum_{0 <= i <= m - |u|}surj_{v-x}(i, v) C(26 - x, v - x) である。 // ただし surj_s(a, b) は a 点集合から b 点集合への写像であって固定した s 点集合が像の部分集合であるものの個数である。 // -> 部分問題に分ける際、S の真の prefix を考慮していなかった。 fn main() { input! { n: usize, m: usize, k: usize, s: chars, } let (fac, invfac) = fact_init(27); // \sum_{0 <= i <= a} surj_s(i, b) let surj_sub_sum = |s: usize, a: usize, b: usize| { let mut tot = MInt::new(0); for i in 0..=s { let tmp = fac[s] * invfac[i] * invfac[s - i]; // 1 + ... + (b-i)^a let tmp = tmp * if b - i == 1 { MInt::new(a as i64 + 1) } else if b == i { MInt::new(1) } else { let t = MInt::new((b - i) as i64); (t.pow(a as i64 + 1) - 1) * fac[b - i - 2] * invfac[b - i - 1] }; if i % 2 == 0 { tot += tmp; } else { tot -= tmp; } } tot }; let mut memo = HashMap::new(); let mut calc = |f: [usize; 26], fsum: usize, k: usize, m: usize| { let mut tot = MInt::new(0); let mut x = 0; for i in 0..26 { if f[i] > 0 { x += 1; } } let key = (x, fsum, m); if let Some(&val) = memo.get(&key) { return val; } tot += (MInt::new(26).pow((m - fsum) as i64 + 1) - 1) * MInt::new(25).inv(); for v in x..k { tot -= surj_sub_sum(v - x, m - fsum, v) * fac[26 - x] * invfac[26 - v] * invfac[v - x]; } memo.insert(key, tot); tot }; let mut f = vec![[0; 26]; n + 1]; for i in 0..n { let idx = (s[i] as u8 - b'a') as usize; f[i + 1] = f[i]; f[i + 1][idx] += 1; } let mut tot = MInt::new(0); for i in 0..n { let idx = (s[i] as u8 - b'a') as usize; for j in 0..idx { let mut g = f[i]; g[j] += 1; tot += calc(g, i + 1, k, m); } } for i in 1..n { tot += calc(f[i], i, k, i); } println!("{}", tot); }