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:heavy_check_mark: setfunc/transposed_sps_composition.hpp

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Code

#include "setfunc/subset_convolution.hpp"

// for fixed sps s, consider linear map F:a->b = subset-conv(a,s)
// given x, calculate transpose(F)(x)
template <typename mint, int LIM>
vc<mint> transposed_subset_convolution(vc<mint> s, vc<mint> x) {
  /*
  sum_{j}x_jb_j = sum_{i subset j}x_ja_is_{j-i} = sum_{i}y_ia_i.
  y_i = sum_{j supset i}x_js_{j-i}
  (rev y)_i = sum_{j subset i}(rev x)_js_{i-j}
  y = rev(conv(rev x), s)
  */
  reverse(all(x));
  x = subset_convolution<mint, LIM>(x, s);
  reverse(all(x));
  return x;
}

// for fixed sps s s.t. s[0] == 0.
// consider linear map F:f->t=f(s) for egf f.
// given x, calcuate transpose(F)(x)
// equivalent: calculate sum_i x_i(s^k/k!)_i for k=0,1,...,N
template <typename mint, int LIM>
vc<mint> transposed_sps_composition_egf(vc<mint>& s, vc<mint> x) {
  const int N = topbit(len(s));
  assert(len(s) == (1 << N) && len(x) == (1 << N) && s[0] == mint(0));
  vc<mint> y(N + 1);
  y[0] = x[0];
  auto& dp = x;
  FOR(i, N) {
    vc<mint> newdp(1 << (N - 1 - i));
    FOR(j, N - i) {
      vc<mint> a = {s.begin() + (1 << j), s.begin() + (2 << j)};
      vc<mint> b = {dp.begin() + (1 << j), dp.begin() + (2 << j)};
      b = transposed_subset_convolution<mint, LIM>(a, b);
      FOR(k, len(b)) newdp[k] += b[k];
    }
    swap(dp, newdp);
    y[1 + i] = dp[0];
  }
  return y;
}
#line 2 "setfunc/subset_convolution.hpp"

#line 2 "setfunc/ranked_zeta.hpp"

template <typename T, int LIM>
vc<array<T, LIM + 1>> ranked_zeta(const vc<T>& f) {
  int n = topbit(len(f));
  assert(n <= LIM);
  assert(len(f) == 1 << n);
  vc<array<T, LIM + 1>> Rf(1 << n);
  for (int s = 0; s < (1 << n); ++s) Rf[s][popcnt(s)] = f[s];
  for (int i = 0; i < n; ++i) {
    int w = 1 << i;
    for (int p = 0; p < (1 << n); p += 2 * w) {
      for (int s = p; s < p + w; ++s) {
        int t = s | 1 << i;
        for (int d = 0; d <= n; ++d) Rf[t][d] += Rf[s][d];
      }
    }
  }
  return Rf;
}

template <typename T, int LIM>
vc<T> ranked_mobius(vc<array<T, LIM + 1>>& Rf) {
  int n = topbit(len(Rf));
  assert(len(Rf) == 1 << n);
  for (int i = 0; i < n; ++i) {
    int w = 1 << i;
    for (int p = 0; p < (1 << n); p += 2 * w) {
      for (int s = p; s < p + w; ++s) {
        int t = s | 1 << i;
        for (int d = 0; d <= n; ++d) Rf[t][d] -= Rf[s][d];
      }
    }
  }
  vc<T> f(1 << n);
  for (int s = 0; s < (1 << n); ++s) f[s] = Rf[s][popcnt(s)];
  return f;
}
#line 4 "setfunc/subset_convolution.hpp"

template <typename T, int LIM = 20>
vc<T> subset_convolution_square(const vc<T>& A) {
  auto RA = ranked_zeta<T, LIM>(A);
  int n = topbit(len(RA));
  FOR(s, len(RA)) {
    auto& f = RA[s];
    FOR_R(d, n + 1) {
      T x = 0;
      FOR(i, d + 1) x += f[i] * f[d - i];
      f[d] = x;
    }
  }
  return ranked_mobius<T, LIM>(RA);
}

template <typename T, int LIM = 20>
vc<T> subset_convolution(const vc<T>& A, const vc<T>& B) {
  if (A == B) return subset_convolution_square(A);
  auto RA = ranked_zeta<T, LIM>(A);
  auto RB = ranked_zeta<T, LIM>(B);
  int n = topbit(len(RA));
  FOR(s, len(RA)) {
    auto &f = RA[s], &g = RB[s];
    FOR_R(d, n + 1) {
      T x = 0;
      FOR(i, d + 1) x += f[i] * g[d - i];
      f[d] = x;
    }
  }
  return ranked_mobius<T, LIM>(RA);
}
#line 2 "setfunc/transposed_sps_composition.hpp"

// for fixed sps s, consider linear map F:a->b = subset-conv(a,s)
// given x, calculate transpose(F)(x)
template <typename mint, int LIM>
vc<mint> transposed_subset_convolution(vc<mint> s, vc<mint> x) {
  /*
  sum_{j}x_jb_j = sum_{i subset j}x_ja_is_{j-i} = sum_{i}y_ia_i.
  y_i = sum_{j supset i}x_js_{j-i}
  (rev y)_i = sum_{j subset i}(rev x)_js_{i-j}
  y = rev(conv(rev x), s)
  */
  reverse(all(x));
  x = subset_convolution<mint, LIM>(x, s);
  reverse(all(x));
  return x;
}

// for fixed sps s s.t. s[0] == 0.
// consider linear map F:f->t=f(s) for egf f.
// given x, calcuate transpose(F)(x)
// equivalent: calculate sum_i x_i(s^k/k!)_i for k=0,1,...,N
template <typename mint, int LIM>
vc<mint> transposed_sps_composition_egf(vc<mint>& s, vc<mint> x) {
  const int N = topbit(len(s));
  assert(len(s) == (1 << N) && len(x) == (1 << N) && s[0] == mint(0));
  vc<mint> y(N + 1);
  y[0] = x[0];
  auto& dp = x;
  FOR(i, N) {
    vc<mint> newdp(1 << (N - 1 - i));
    FOR(j, N - i) {
      vc<mint> a = {s.begin() + (1 << j), s.begin() + (2 << j)};
      vc<mint> b = {dp.begin() + (1 << j), dp.begin() + (2 << j)};
      b = transposed_subset_convolution<mint, LIM>(a, b);
      FOR(k, len(b)) newdp[k] += b[k];
    }
    swap(dp, newdp);
    y[1 + i] = dp[0];
  }
  return y;
}
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