gh-35173: Speedup of Poset characteristic polynomial
### 📚 Description
Add special method for Moebius function computation with minimal
element, add cached_method to HasseDiagram.bottom, change to sum over
bottom_moebius_function
This drastically reduces the time required to compute characteristic
polynomials. On commit `fbb412787f`, we see
sage: P = posets.SetPartitions(7)
sage: %time P.characteristic_polynomial()
CPU times: user 42.2 s, sys: 475 ms, total: 42.7 s
Wall time: 45.2 s
q^6 - 21*q^5 + 175*q^4 - 735*q^3 + 1624*q^2 - 1764*q + 720
With this commit we see:
sage: P = posets.SetPartitions(7)
sage: %time P.characteristic_polynomial()
CPU times: user 62.2 ms, sys: 1.83 ms, total: 64 ms
Wall time: 63.1 ms
q^6 - 21*q^5 + 175*q^4 - 735*q^3 + 1624*q^2 - 1764*q + 720
Fixes #34971
### 📝 Checklist
- [x] I have made sure that the title is self-explanatory and the
description concisely explains the PR.
- [x] I have linked an issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation accordingly.
### ⌛ Dependencies
None
@tscrim is a suggested reviewer.
URL: https://github.com/sagemath/sage/pull/35173
Reported by: trevorkarn
Reviewer(s): Travis Scrimshaw, trevorkarn
