Problem:
Score : 400 points

Problem Statement

In this problem, we only consider strings consisting of lowercase English letters.

Strings s and t are said to be isomorphic when the following conditions are satisfied:

|s| = |t| holds.

For every pair i, j, one of the following holds:

s_i = s_j and t_i = t_j.

s_i \neq s_j and t_i \neq t_j.

For example, abcac and zyxzx are isomorphic, while abcac and ppppp are not.

A string s is said to be in normal form when the following condition is satisfied:

For every string t that is isomorphic to s, s \leq t holds. Here \leq denotes lexicographic comparison.

For example, abcac is in normal form, but zyxzx is not since it is isomorphic to abcac, which is lexicographically smaller than zyxzx.

You are given an integer N.
Print all strings of length N that are in normal form, in lexicographically ascending order.

Constraints

1 \leq N \leq 10

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N

Output

Assume that there are K strings of length N that are in normal form: w_1, \ldots, w_K in lexicographical order.
Output should be in the following format:

w_1
:
w_K

Sample Input 1

1

Sample Output 1

a

Sample Input 2

2

Sample Output 2

aa
ab

Solution in Rust (1.15.1) (judge verdict: Wrong Answer)

// The main code is at the very bottom.

#[allow(unused_imports)]
#[macro_use]
pub mod spella {
  pub mod byte {
    pub use self::byte_char::*;
    pub use self::byte_str::*;
    pub use self::byte_string::*;
    pub use self::from_byte_str::*;

    mod byte_char {
      use std::fmt::{self, Debug, Display, Formatter};

      #[derive(Clone, Copy, Default, PartialEq, Eq, PartialOrd, Ord, Hash)]
      pub struct ByteChar(pub u8);

      impl Debug for ByteChar {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          write!(f, "b'{}'", self.0 as char)
        }
      }

      impl Display for ByteChar {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          write!(f, "{}", self.0 as char)
        }
      }
    }

    mod byte_str {
      use spella::byte::{ByteChar, ByteString};
      use std::fmt::{self, Debug, Display, Formatter};
      use std::ops::{Deref, DerefMut};

      #[derive(PartialEq, Eq, PartialOrd, Ord, Hash)]
      pub struct ByteStr([ByteChar]);

      macro_rules! cast {
        (mut $x:expr, $($T:ty)=>*) => {
          unsafe { &mut *($x $(as *mut $T)*) }
        };
        ($x:expr, $($T:ty)=>*) => {
          unsafe { &*($x $(as *const $T)*) }
        };
      }

      impl ByteStr {
        pub fn from_bytes(s: &[u8]) -> &Self {
          cast!(s, [u8] => [ByteChar] => ByteStr)
        }

        pub fn from_bytes_mut(s: &mut [u8]) -> &mut Self {
          cast!(mut s, [u8] => [ByteChar] => ByteStr)
        }

        pub fn from_byte_chars(s: &[ByteChar]) -> &Self {
          cast!(s, [ByteChar] => ByteStr)
        }

        pub fn from_byte_chars_mut(s: &mut [ByteChar]) -> &mut Self {
          cast!(mut s, [ByteChar] => ByteStr)
        }

        pub fn as_byte_chars(&self) -> &[ByteChar] {
          &self.0
        }

        pub fn as_byte_chars_mut(&mut self) -> &mut [ByteChar] {
          &mut self.0
        }

        pub fn as_bytes(&self) -> &[u8] {
          cast!(self, ByteStr => [ByteChar] => [u8])
        }

        pub fn as_bytes_mut(&mut self) -> &mut [u8] {
          cast!(mut self, ByteStr => [ByteChar] => [u8])
        }
      }

      impl ToOwned for ByteStr {
        type Owned = ByteString;

        fn to_owned(&self) -> ByteString {
          ByteString::from(self.0.to_owned())
        }
      }

      impl Debug for ByteStr {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          write!(f, "b\"")?;

          for &c in &self.0 {
            write!(f, "{}", c)?;
          }

          write!(f, "\"")
        }
      }

      impl Display for ByteStr {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          for &c in &self.0 {
            write!(f, "{}", c)?;
          }

          Ok(())
        }
      }

      impl Deref for ByteStr {
        type Target = [ByteChar];

        fn deref(&self) -> &[ByteChar] {
          self.as_byte_chars()
        }
      }

      impl DerefMut for ByteStr {
        fn deref_mut(&mut self) -> &mut [ByteChar] {
          self.as_byte_chars_mut()
        }
      }
    }

    mod byte_string {
      use spella::byte::{ByteChar, ByteStr};

      use std::borrow::{Borrow, BorrowMut};
      use std::fmt::{self, Debug, Display, Formatter};
      use std::ops::{Deref, DerefMut};

      #[derive(Clone, Default, PartialEq, Eq, PartialOrd, Ord, Hash)]
      pub struct ByteString(Vec<ByteChar>);

      impl ByteString {
        pub fn into_byte_chars(self) -> Vec<ByteChar> {
          self.0
        }

        pub fn as_byte_str(&self) -> &ByteStr {
          ByteStr::from_byte_chars(&self.0)
        }

        pub fn as_mut_byte_str(&mut self) -> &mut ByteStr {
          ByteStr::from_byte_chars_mut(&mut self.0)
        }
      }

      impl From<Vec<ByteChar>> for ByteString {
        fn from(s: Vec<ByteChar>) -> ByteString {
          ByteString(s)
        }
      }

      impl Borrow<ByteStr> for ByteString {
        fn borrow(&self) -> &ByteStr {
          self.as_byte_str()
        }
      }

      impl BorrowMut<ByteStr> for ByteString {
        fn borrow_mut(&mut self) -> &mut ByteStr {
          self.as_mut_byte_str()
        }
      }

      impl Debug for ByteString {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          Debug::fmt(self.as_byte_str(), f)
        }
      }

      impl Display for ByteString {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          Display::fmt(self.as_byte_str(), f)
        }
      }

      impl Deref for ByteString {
        type Target = ByteStr;

        fn deref(&self) -> &ByteStr {
          ByteStr::from_byte_chars(&self.0)
        }
      }

      impl DerefMut for ByteString {
        fn deref_mut(&mut self) -> &mut ByteStr {
          ByteStr::from_byte_chars_mut(&mut self.0)
        }
      }
    }

    mod from_byte_str {
      use spella::byte::{ByteChar, ByteStr, ByteString};

      use std::error::Error;
      use std::fmt::{self, Debug, Display, Formatter};
      use std::str::{self, FromStr, Utf8Error};

      pub trait FromByteStr: Sized {
        type Err;

        fn from_byte_str(s: &ByteStr) -> Result<Self, Self::Err>;
      }

      macro_rules! fn_description {
        () => {
          fn description(&self) -> &str {
            "description() is deprecated; use Display"
          }
        };
      }

      #[derive(Debug)]
      pub struct ParseByteCharError(ParseByteCharErrorKind);

      #[derive(Debug)]
      enum ParseByteCharErrorKind {
        EmptyByteStr,
        TooManyByteChars,
      }

      impl Display for ParseByteCharError {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          use self::ParseByteCharErrorKind::*;

          f.write_str(match self.0 {
            EmptyByteStr => "empty `ByteStr`",
            TooManyByteChars => "too many `ByteChar`s",
          })
        }
      }

      impl Error for ParseByteCharError {
        fn_description! {}
      }

      impl FromByteStr for ByteChar {
        type Err = ParseByteCharError;

        fn from_byte_str(s: &ByteStr) -> Result<Self, Self::Err> {
          use self::ParseByteCharErrorKind::*;

          match s.len() {
            1 => Ok(unsafe { *s.get_unchecked(0) }),
            0 => Err(ParseByteCharError(EmptyByteStr)),
            _ => Err(ParseByteCharError(TooManyByteChars)),
          }
        }
      }

      #[derive(Debug)]
      pub enum ParseByteStringError {}

      impl Display for ParseByteStringError {
        fn fmt(&self, _: &mut Formatter) -> fmt::Result {
          match *self {}
        }
      }

      impl Error for ParseByteStringError {
        fn_description! {}
      }

      impl FromByteStr for ByteString {
        type Err = ParseByteStringError;

        fn from_byte_str(s: &ByteStr) -> Result<Self, Self::Err> {
          Ok(ByteString::from(s.to_vec()))
        }
      }

      pub struct ParseFromStrError<T: FromStr>(ParseFromStrErrorKind<T>);

      enum ParseFromStrErrorKind<T: FromStr> {
        Utf8Error(Utf8Error),
        FromStrError(T::Err),
      }

      impl<T: FromStr> Debug for ParseFromStrError<T>
      where
        T::Err: Debug,
      {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          use self::ParseFromStrErrorKind::*;

          match self.0 {
            Utf8Error(ref err) => f.debug_tuple("Utf8Error").field(err).finish(),
            FromStrError(ref err) => f.debug_tuple("FromStrError").field(err).finish(),
          }
        }
      }

      impl<T: FromStr> Display for ParseFromStrError<T>
      where
        T::Err: Display,
      {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          use self::ParseFromStrErrorKind::*;

          match self.0 {
            Utf8Error(ref err) => write!(f, "{}", err),
            FromStrError(ref err) => write!(f, "{}", err),
          }
        }
      }

      impl<T: FromStr> Error for ParseFromStrError<T>
      where
        T::Err: Debug + Display,
      {
        fn_description! {}
      }

      impl<T: FromStr> FromByteStr for T {
        type Err = ParseFromStrError<T>;

        fn from_byte_str(s: &ByteStr) -> Result<T, Self::Err> {
          use self::ParseFromStrErrorKind::*;

          str::from_utf8(s.as_bytes())
            .map_err(|e| ParseFromStrError(Utf8Error(e)))
            .and_then(|s| s.parse().map_err(|e| ParseFromStrError(FromStrError(e))))
        }
      }
    }
  }

  pub mod io {
    pub use self::scanner::*;

    mod scanner {
      use spella::byte::{ByteStr, FromByteStr};

      use std::io::{self, BufRead};

      #[derive(Debug)]
      pub struct Scanner<R> {
        reader: R,
        buf: Vec<u8>,
        pos: usize,
      }

      const INITIAL_CAPACITY: usize = 32;

      impl<R: BufRead> Scanner<R> {
        pub fn new(reader: R) -> Self {
          Scanner {
            reader: reader,
            buf: Vec::with_capacity(INITIAL_CAPACITY),
            pos: 0,
          }
        }

        pub fn next<T: FromByteStr>(&mut self) -> io::Result<Result<T, T::Err>> {
          self.next_byte_str().map(T::from_byte_str)
        }

        pub fn next_byte_str(&mut self) -> io::Result<&ByteStr> {
          if self.buf.is_empty() {
            self.read_line()?;
          }

          loop {
            match self.buf.get(self.pos) {
              Some(&b' ') => self.pos += 1,
              Some(&b'\n') => self.read_line()?,
              Some(_) => break,
              None => return Err(io::Error::from(io::ErrorKind::UnexpectedEof)),
            }
          }

          let start = self.pos;
          self.pos += 1;

          loop {
            match self.buf.get(self.pos) {
              Some(&b' ') | Some(&b'\n') | None => break,
              Some(_) => self.pos += 1,
            }
          }

          Ok(ByteStr::from_bytes(&self.buf[start..self.pos]))
        }

        fn read_line(&mut self) -> io::Result<()> {
          self.buf.clear();
          self.pos = 0;
          self.reader.read_until(b'\n', &mut self.buf)?;

          Ok(())
        }
      }
    }
  }
}

pub mod stopgap {
  use std::fmt::{self, Debug, Display, Formatter};
  use std::hash::{Hash, Hasher};
  use std::iter::{Product, Sum};
  use std::marker::PhantomData;
  use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign};

  pub trait Constant<T> {
    fn get() -> T;
  }

  pub enum _1000000007 {}

  impl Constant<u32> for _1000000007 {
    fn get() -> u32 {
      1000000007
    }
  }

  pub struct ModU32<Mod> {
    _phantom: PhantomData<fn() -> Mod>,
    repr: u32,
  }

  impl<Mod: Constant<u32>> Clone for ModU32<Mod> {
    fn clone(&self) -> Self {
      Self::from_repr(self.repr)
    }
  }

  impl<Mod: Constant<u32>> Copy for ModU32<Mod> {}

  impl<Mod: Constant<u32>> PartialEq for ModU32<Mod> {
    fn eq(&self, other: &Self) -> bool {
      self.repr.eq(&other.repr)
    }
  }

  impl<Mod: Constant<u32>> Eq for ModU32<Mod> {}

  impl<Mod: Constant<u32>> Default for ModU32<Mod> {
    fn default() -> Self {
      ModU32::from_repr(u32::default())
    }
  }

  impl<Mod: Constant<u32>> Hash for ModU32<Mod> {
    fn hash<H>(&self, state: &mut H)
    where
      H: Hasher,
    {
      self.repr.hash(state);
    }
  }

  impl<Mod: Constant<u32>> Debug for ModU32<Mod> {
    fn fmt(&self, f: &mut Formatter) -> fmt::Result {
      Debug::fmt(&self.repr, f)
    }
  }

  impl<Mod: Constant<u32>> Display for ModU32<Mod> {
    fn fmt(&self, f: &mut Formatter) -> fmt::Result {
      Display::fmt(&self.repr, f)
    }
  }

  impl<Mod: Constant<u32>> ModU32<Mod> {
    pub fn new(value: u32) -> Self {
      Self::from_repr(value % Mod::get())
    }

    fn from_repr(repr: u32) -> Self {
      debug_assert!(repr < Mod::get());
      ModU32 {
        _phantom: PhantomData,
        repr: repr,
      }
    }

    pub fn repr(self) -> u32 {
      self.repr
    }

    pub fn inverse(self) -> Self {
      let mut a = self.repr as i64;
      let mut b = Mod::get() as i64;
      let mut u = 1i64;
      let mut v = 0i64;
      while b != 0 {
        let t = a / b;
        a -= t * b;
        ::std::mem::swap(&mut a, &mut b);
        u -= t * v;
        ::std::mem::swap(&mut u, &mut v);
      }
      u %= Mod::get() as i64;
      if u < 0 {
        u += Mod::get() as i64;
      }
      Self::from_repr(u as u32)
    }

    pub fn pow(mut self, mut n: u32) -> Self {
      let mut acc = ModU32::from_repr(1);
      while n > 0 {
        if n % 2 == 1 {
          acc *= self;
        }
        self *= self;
        n /= 2;
      }
      acc
    }
  }

  impl<Mod: Constant<u32>> Add for ModU32<Mod> {
    type Output = Self;

    fn add(mut self, rhs: Self) -> Self {
      self += rhs;
      self
    }
  }

  impl<Mod: Constant<u32>> AddAssign for ModU32<Mod> {
    fn add_assign(&mut self, rhs: Self) {
      self.repr += rhs.repr;
      if self.repr >= Mod::get() {
        self.repr -= Mod::get();
      }
    }
  }

  impl<Mod: Constant<u32>> Sub for ModU32<Mod> {
    type Output = Self;

    fn sub(mut self, rhs: Self) -> Self {
      self -= rhs;
      self
    }
  }

  impl<Mod: Constant<u32>> SubAssign for ModU32<Mod> {
    fn sub_assign(&mut self, rhs: Self) {
      if self.repr < rhs.repr {
        self.repr += Mod::get();
      }
      self.repr -= rhs.repr;
    }
  }

  impl<Mod: Constant<u32>> Mul for ModU32<Mod> {
    type Output = Self;

    fn mul(self, rhs: Self) -> Self {
      Self::from_repr((self.repr as u64 * rhs.repr as u64 % Mod::get() as u64) as u32)
    }
  }

  impl<Mod: Constant<u32>> MulAssign for ModU32<Mod> {
    fn mul_assign(&mut self, rhs: Self) {
      *self = *self * rhs;
    }
  }

  impl<Mod: Constant<u32>> Div for ModU32<Mod> {
    type Output = Self;

    fn div(self, rhs: Self) -> Self {
      self * rhs.inverse()
    }
  }

  impl<Mod: Constant<u32>> DivAssign for ModU32<Mod> {
    fn div_assign(&mut self, rhs: Self) {
      *self *= rhs.inverse();
    }
  }

  impl<Mod: Constant<u32>> Sum for ModU32<Mod> {
    fn sum<I>(iter: I) -> Self
    where
      I: Iterator<Item = Self>,
    {
      let mut acc = ModU32::from_repr(0);
      for x in iter {
        acc += x;
      }
      acc
    }
  }

  impl<Mod: Constant<u32>> Product for ModU32<Mod> {
    fn product<I>(iter: I) -> Self
    where
      I: Iterator<Item = Self>,
    {
      let mut acc = ModU32::from_repr(1);
      for x in iter {
        acc *= x;
      }
      acc
    }
  }

  pub fn mod_pow<Mod: Constant<u32>>(base: u32, n: u32) -> ModU32<Mod> {
    ModU32::new(base).pow(n)
  }

  pub fn mod_binomial<Mod: Constant<u32>>(mut n: u32, mut k: u32) -> ModU32<Mod> {
    let mut acc = ModU32::from_repr(1);
    while k != 0 {
      {
        let n = n % Mod::get();
        let k = k % Mod::get();
        if n < k {
          return ModU32::from_repr(0);
        }
        acc *= (n - k + 1..n + 1).map(ModU32::from_repr).product();
        acc /= (1..k + 1).map(ModU32::from_repr).product();
      }
      n /= Mod::get();
      k /= Mod::get();
    }
    acc
  }
}

#[allow(unused_imports)]
use stopgap::*;

mod prelude {
  pub use spella::byte::{ByteChar, ByteStr, ByteString};
  pub use std::collections::*;
  pub use std::io::prelude::*;
  pub use std::iter::FromIterator;
  pub use std::marker::PhantomData;
  pub use std::num::Wrapping;
  pub use std::ops::{Range, RangeFrom, RangeTo};
  pub use std::{cell, cmp, f64, i32, i64, isize, iter, mem, rc, str, time, u32, u64, usize};
}

use prelude::*;

const CUSTOM_STACK_SIZE_MEBIBYTES: Option<usize> = None;

fn main() {
  fn exec_solver() {
    let stdin = std::io::stdin();
    let stdout = std::io::stdout();
    #[cfg(not(debug_assertions))]
    let mut writer = std::io::BufWriter::new(stdout.lock());
    #[cfg(debug_assertions)]
    let mut writer = stdout.lock();
    solve(stdin.lock(), &mut writer);
    writer.flush().unwrap();
  }

  if let Some(stack_size_mebibytes) = CUSTOM_STACK_SIZE_MEBIBYTES {
    std::thread::Builder::new()
      .name("exec_solver".to_owned())
      .stack_size(stack_size_mebibytes * 1024 * 1024)
      .spawn(exec_solver)
      .unwrap()
      .join()
      .unwrap();
  } else {
    exec_solver();
  }
}

fn solve<R: BufRead, W: Write>(reader: R, mut writer: W) {
  let mut _scanner = spella::io::Scanner::new(reader);

  #[allow(unused_macros)]
  macro_rules! scan {
    ($T:ty) => {
      _scanner.next::<$T>().unwrap().unwrap()
    };
    ($($T:ty),+) => {
      ($(scan!($T)),+)
    };
    ($($T:ty),+; $n:expr $(; $m:expr)*) => {{
      (0..$n).map(|_| scan!($($T),+ $(; $m)*)).collect::<Vec<_>>()
    }};
  }

  #[allow(unused_macros)]
  macro_rules! scan_iter {
    ($($T:ty),+; $n:expr) => {
      (0..$n).map(|_| scan!($($T),+))
    };
  }

  #[allow(unused_macros)]
  macro_rules! print {
    ($fmt:expr) => {
      write!(writer, $fmt).unwrap()
    };
    ($fmt:expr, $($arg:tt)*) => {
      write!(writer, $fmt, $($arg)*).unwrap()
    };
  }

  #[allow(unused_macros)]
  macro_rules! println {
    ($fmt:expr) => {
      writeln!(writer, $fmt).unwrap()
    };
    ($fmt:expr, $($arg:tt)*) => {
      writeln!(writer, $fmt, $($arg)*).unwrap()
    };
  }

  #[allow(unused_macros)]
  macro_rules! eprint {
    ($fmt:expr) => {
      #[cfg(debug_assertions)]
      write!(std::io::stderr(), $fmt).unwrap()
    };
    ($fmt:expr, $($arg:tt)*) => {
      #[cfg(debug_assertions)]
      write!(std::io::stderr(), $fmt, $($arg)*).unwrap()
    };
  }

  #[allow(unused_macros)]
  macro_rules! eprintln {
    ($fmt:expr) => {
      #[cfg(debug_assertions)]
      writeln!(std::io::stderr(), $fmt).unwrap()
    };
    ($fmt:expr, $($arg:tt)*) => {
      #[cfg(debug_assertions)]
      writeln!(std::io::stderr(), $fmt, $($arg)*).unwrap()
    };
  }

  #[allow(unused_macros)]
  macro_rules! dbg {
    ($($x:expr),+) => {{
      eprintln!(concat!("[{}:{}] ", $(stringify!($x), " = {:?}; "),+), file!(), line!(), $($x),+);
    }};
  }

  let n = scan!(usize);

  fn f(c: ByteChar, n: usize) -> Vec<VecDeque<ByteChar>> {
    if n == 1 {
      vec![vec![c].into()]
    } else {
      let mut a = f(c, n - 1);
      let mut b = f(ByteChar(c.0 + 1), n - 1);
      a.append(&mut b);
      for s in &mut a {
        s.push_front(c);
      }
      a
    }
  }

  let mut ans = f(ByteChar(b'a'), n);
  ans.sort();
  for s in ans {
    for c in s {
      print!("{}", c);
    }
    println!("");
  }
}