Task: Partition function P



C#: using System;

class Program {

    const long Lm = (long)1e18;
    const string Fm = "D18";

    
    struct LI { public long lo, ml, mh, hi, tp; }

    static void inc(ref LI d, LI s) { 
        if ((d.lo += s.lo) >= Lm) { d.ml++; d.lo -= Lm; }
        if ((d.ml += s.ml) >= Lm) { d.mh++; d.ml -= Lm; }
        if ((d.mh += s.mh) >= Lm) { d.hi++; d.mh -= Lm; }
        if ((d.hi += s.hi) >= Lm) { d.tp++; d.hi -= Lm; }
        d.tp += s.tp;
    }
 
    static void dec(ref LI d, LI s) { 
        if ((d.lo -= s.lo) < 0) { d.ml--; d.lo += Lm; }
        if ((d.ml -= s.ml) < 0) { d.mh--; d.ml += Lm; }
        if ((d.mh -= s.mh) < 0) { d.hi--; d.mh += Lm; }
        if ((d.hi -= s.hi) < 0) { d.tp--; d.hi += Lm; }
        d.tp -= s.tp;
    }

    static LI set(long s) { LI d;
      d.lo = s; d.ml = d.mh = d.hi = d.tp = 0; return d; }

  static string fmt(LI x) { 
    if (x.tp > 0) return x.tp.ToString() + x.hi.ToString(Fm) + x.mh.ToString(Fm) + x.ml.ToString(Fm) + x.lo.ToString(Fm);
    if (x.hi > 0) return x.hi.ToString() + x.mh.ToString(Fm) + x.ml.ToString(Fm) + x.lo.ToString(Fm);
    if (x.mh > 0) return x.mh.ToString() + x.ml.ToString(Fm) + x.lo.ToString(Fm);
    if (x.ml > 0) return x.ml.ToString() + x.lo.ToString(Fm);
    return x.lo.ToString();
  }

  static LI partcount(int n) {
    var P = new LI[n + 1]; P[0] = set(1);
    for (int i = 1; i <= n; i++) {
      int k = 0, d = -2, j = i;
      LI x = set(0);
      while (true) {
        if ((j -= (d += 3) -k) >= 0) inc(ref x, P[j]); else break;
        if ((j -= ++k)         >= 0) inc(ref x, P[j]); else break;
        if ((j -= (d += 3) -k) >= 0) dec(ref x, P[j]); else break;
        if ((j -= ++k)         >= 0) dec(ref x, P[j]); else break;
      }
      P[i] = x;
    }
    return P[n];
  }

  static void Main(string[] args) {
    var sw = System.Diagnostics.Stopwatch.StartNew ();
    var res = partcount(6666); sw.Stop();
    Console.Write("{0}  {1} ms", fmt(res), sw.Elapsed.TotalMilliseconds);
  }
}





Python: from itertools import islice

def posd():
    "diff between position numbers. 1, 2, 3... interleaved with  3, 5, 7..."
    count, odd = 1, 3
    while True:
        yield count
        yield odd
        count, odd = count + 1, odd + 2

def pos_gen():
    "position numbers. 1 3 2 5 7 4 9 ..."
    val = 1
    diff = posd()
    while True:
        yield val
        val += next(diff)
                
def plus_minus():
    "yield (list_offset, sign) or zero for Partition calc"
    n, sign = 0, [1, 1]
    p_gen = pos_gen()
    out_on = next(p_gen)
    while True:
        n += 1
        if n == out_on:
            next_sign = sign.pop(0)
            if not sign:
                sign = [-next_sign] * 2
            yield -n, next_sign
            out_on = next(p_gen)
        else:
            yield 0
            
def part(n):
    "Partition numbers"
    p = [1]
    p_m = plus_minus()
    mods = []
    for _ in range(n):
        next_plus_minus = next(p_m)
        if next_plus_minus:
            mods.append(next_plus_minus)
        p.append(sum(p[offset] * sign for offset, sign in mods))
    return p[-1]
        
print("(Intermediaries):")
print("    posd:", list(islice(posd(), 10)))
print("    pos_gen:", list(islice(pos_gen(), 10)))
print("    plus_minus:", list(islice(plus_minus(), 15)))
print("\nPartitions:", [part(x) for x in range(15)])