#[allow(unused_imports)] use std::cmp::*; #[allow(unused_imports)] use std::collections::*; 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 = 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) } fn perm_comp(a: &[usize], b: &[usize]) -> Vec { let n = a.len(); assert_eq!(b.len(), n); let mut c = vec![0; n]; for i in 0..n { c[i] = a[b[i]]; } c } fn perm_inv(a: &[usize]) -> Vec { let n = a.len(); let mut c = vec![0; n]; for i in 0..n { c[a[i]] = i; } c } // Returns the least index of elements that are modified, wrapped with Some. // If the entire array is reversed, it returns None instead. // v's elements must be pairwise distinct. fn next_permutation(v: &mut [T]) -> Option { let mut tail_dec: usize = 1; let n = v.len(); while tail_dec < n { if v[n - tail_dec - 1] > v[n - tail_dec] { tail_dec += 1; } else { break; } } // v[n - tail_dec .. n] is strictly decreasing if tail_dec < n { let x = n - tail_dec - 1; let mut y = n; { let pivot = &v[x]; for i in (n - tail_dec..n).rev() { if v[i] > *pivot { y = i; break; } } assert!(y < n); } v.swap(x, y); } v[n - tail_dec..].reverse(); if tail_dec < n { Some(n - tail_dec - 1) } else { None } } // https://yukicoder.me/problems/no/2384 (5) // F は S_N の自己同型群である。 // (i) N = 2 のとき、F の要素は恒等写像のみ。これが条件を満たすので答えは 1 である。 // (ii) N != 2, 6 のとき、これは S_N と同型であり、要素は共役作用のみである。 // 条件を満たす F の要素と 1 から K までをなんらかの円環シフトする置換は 1:1 に対応するので、答えは (N-K)!K である。 // (iii) N = 6 のとき、the outer automorphism が存在する。 fn main() { let n: usize = get(); let k: usize = get(); if n == 2 { println!("1"); return; } let (fac, _invfac) = fact_init(n + 1); if n != 6 { println!("{}", fac[n - k] * k as i64); return; } let mut tot = fac[n - k] * k as i64; // One representative of the outer automorphism. // https://mathstoshare.com/2019/12/16/the-outer-automorphism-of-s6/ let outer = vec![ vec![1, 0, 3, 2, 5, 4], vec![2, 4, 0, 5, 1, 3], vec![4, 5, 3, 2, 0, 1], vec![2, 3, 0, 1, 5, 4], vec![5, 4, 3, 2, 1, 0], ]; let mut f: Vec<_> = (0..6).collect(); for i in 0..k - 1 { f = perm_comp(&outer[i], &f); } let mut p: Vec<_> = (0..6).collect(); loop { let pinv = perm_inv(&p); let q = perm_comp(&pinv, &perm_comp(&f, &p)); if (0..k).all(|i| q[i] == (i + 1) % k) { tot += 1; } if let None = next_permutation(&mut p) { break; } } println!("{}", tot); }