TO DO
- thompson NFA finish it simple as possible only support * and .
- regex and partitions with zeroes. Look at this: a*a*a* and string aaa
  The ways to match (each number represents a pattern unit, a*)
  0,0,3
  0,1,2
  1,1,1
  0,2,1
  0,3,0
  1,2,0
  2,1,0
  3,0,0
  Isn't this somehow connected to the partitions of 3 with zeroes?
  If I sort them
  0,0,3
  0,1,2
  1,1,1
  To get the basis partitions, then rearrange I get all of them. It's connected, 
  but possible not deeply connected, but I think it's also connected with 
    the factors (and ways to arrange them) of a number. And possibly the sigma function
    (number of divisors)
- Primes and perfect numbers, sigma(p) = p + 1, and sigma(perfect) = 2p
  Perfect numbers might have some special patterning in the numbers, or in relation to the primes
  In a sense they are like primes of the second order from the point of view of the sum of divisors function
  Sigma_0(p) = p + 1 sigma_1(perfect) = 2p Near relationships between sigma_k(n) and n is something 
  special
- Number theory, expressing n=pq in terms of euler totioent and sum of divisors functions
https://en.wikipedia.org/wiki/Divisor_function#Other_properties_and_identities
And these identities are cool too https://en.wikipedia.org/wiki/Arithmetic_function#Divisor_sum_convolutions
- orders. Hasse diagrams. This stuff is fascinating. (see: https://archive.is/wip/nP1Oh)
- equivalence classes of integers under partitions as determined by binary representation
  - let's do a topolgocial sort by these classes. See NOTES in the 01 directory
  - I want to put this into some graph representation to really see the patterns come out. And some graphs (maybe in grids) with colors and numbers, etc. I need to think about, and I think it will be fun to think about the best way to arrange the data to really let the patterns fly out.
  - second to this investigate the partitions given by and the equivalence classes of numbers in 
    irrational (base phi, base e, etc (c.f. irradix)) radix representations
  - third consider what if we change partition definition, or allow 0s to be included in partitions?
- kth permutation of a string / ordered set. That means we can define any sequence in terms of
S (ordered set or string) and k (the kth permutation) as well as a partial order p.
- question: if you have a box of N pairs of socks, how many socks do you have to draw until you get a match with probability p? (might be related to birthday spacings). Now, what if you throw a spanner in the works with K unmatched (lost / forgotten) half-pairs? Now, what if you throw another spanner in the works with a function Z(S,S) => R0,1 that fuzzily compares socks, so you no longer have an exact (0,1) match but a fuzzy continuum of match-iness, what's the relationship between p, N, and Z?

DOING
- equivalence classes of integers under partitions as determined by binary representation

PARTIALLY DONE 
  PARTIALLY DONE - what's the connection between bit sequences, repeated squarings, reconstructing a number via a sequence of doublings and additions, and deconstructing via the reverse
    - see: commit 58d617d7b50c868a6d2153052b8e95492e04f63f
    - still to do: really come up with THE set of equations and algorithms for all this stuff
    the connections, flesh it all out. I just feel like 
    there's something there to be obtained in the clarity about this.