use std::io::Read; fn get_word() -> String { let stdin = std::io::stdin(); let mut stdin=stdin.lock(); let mut u8b: [u8; 1] = [0]; loop { let mut buf: Vec = Vec::with_capacity(16); loop { let res = stdin.read(&mut u8b); if res.unwrap_or(0) == 0 || u8b[0] <= b' ' { break; } else { buf.push(u8b[0]); } } if buf.len() >= 1 { let ret = String::from_utf8(buf).unwrap(); return ret; } } } fn get() -> T { get_word().parse().ok().unwrap() } /// 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 = 1_000_000_007; 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) } const B: usize = 2; // Depends on MInt.rs // Verified by: https://atcoder.jp/contests/abc199/submissions/22259436 fn squmul(a: &[[MInt; B]], b: &[[MInt; B]]) -> [[MInt; B]; B] { let mut ret = [[MInt::new(0); B]; B]; for i in 0..B { for j in 0..B { for k in 0..B { ret[i][k] += a[i][j] * b[j][k]; } } } ret } fn squpow(a: &[[MInt; B]; B], mut e: i64) -> [[MInt; B]; B] { let mut sum = [[MInt::new(0); B]; B]; for i in 0..B { sum[i][i] = 1.into(); } let mut cur = *a; while e > 0 { if e % 2 == 1 { sum = squmul(&sum, &cur); } cur = squmul(&cur, &cur); e /= 2; } sum } fn calc(n: i32, m: i64, k: i64) -> MInt { if n == 1 { return MInt::new(k - 1).pow(m - 1) * k; } if n == 2 { return MInt::new(k * k + 3 - 3 * k).pow(m - 1) * k * (k - 1); } let mut mat = [[MInt::new(0); 2]; 2]; mat[0][0] += (k - 2) * (k - 2) + k - 1; mat[0][1] += k * (k - 1) * (k - 2) - 3 * (k - 1) * (k - 2) + 2 * (k - 2); mat[1][0] += k * (k - 1) - 3 * (k - 1) + 2; mat[1][1] += k * (k - 1) * (k - 2) - 3 * (k - 1) * (k - 2) + 3 * (k - 2) - 1; let pw = squpow(&mat, m - 1); let mut ans = (pw[0][0] + pw[0][1]) * (k * (k - 1)); ans += (pw[1][0] + pw[1][1]) * (k * (k - 1) * (k - 2)); ans } // https://yukicoder.me/problems/no/1815 (4) // 包除原理。N=3 のときは、列ごとの状態は「すべての色が異なる」か「1 列目と 3 列目が等しい」の 2 通りのみなので、2 次正方行列の行列累乗でできる。 // allocation を少なくする必要あり。 // Complexity: O(k log m) fn main() { let n: i32 = get(); let m: i64 = get(); let k: i64 = get(); let (fac, invfac) = fact_init(k as usize + 2); let mut coef = MInt::new(1); let mut ans = MInt::new(0); for i in 0..k + 1 { let tmp = calc(n, m, k - i); ans += coef * tmp; coef *= -MInt::new(k - i); coef *= invfac[i as usize + 1] * fac[i as usize]; } println!("{}", ans); }