Permutations, Combinations, and the Path of the Arcanist

The usage of arcane glyphs and diagrams to power a wide variety of occult rituals.
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Permutations, Combinations, and the Path of the Arcanist

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Based on the voice chat from 19 February, I wanted to provide some numbers around larger occult circles, particularly with regards to the idea of interaction of effect glyphs.

I'll now need to throw out some assumptions because data do not yet exist:
- Suppose the upper bound for effect glyphs is 3
- Order is important (ie drawing the same glyphs in a different order is different)
- Replacement is allowed (meaning the same glyph can be used more than once in the same circle)
With the 6 glyphs presently implemented, that would give 294 permutations (6*7*7, sevens because 'no glyph in this slot' is an option after the first). If we remove duplication (e.g. distyr, distyr, dorun is equivalent to distyr, dorun) that trims unique permutations down to 180. If instead we say order isn't important (e.g distyr, dorun, uyto is equivalent to uyto, dorun, distyr) then that would trim the unique combinations down to 68. With no replacement and order isn't important we end up with 35 combinations, and 6 of those combinations are 'glyph, no effect glyph, no effect glyph', so those are already defined and only 29 new effects would need to be considered.

With that in mind, consider the introduction of the next glyph:
- 294 permutations increases to 448, for 154 new effects to consider, less the base effect for the new glyph for 153.
- 180 permutations without duplication increases to 294, for 113 new effects to consider.
- 68 combinations disregarding order increases to 98, for 29 new effects to consider.
- 35 combinations disregarding order and duplicates increases to 56, for 20 new effects to consider.

The second marginal glyph (8 glyphs implemented) shows the impact of increasing complexity:
- 448 permutations increases to 648, for 199 new effecs to consider, 45 more than that last marginal glyph.
- 294 permutations without duplication increases to 448, for 153 new effects to consider, 40 more than the last marginal glyph.
- 98 combinations disregarding order increases to 136, for 37 new effecs to consider, 8 more than the last marginal glyph.
- 56 combinations disregarding order and duplicates increases to 84, for 27 new effects to consider, and 7 more than the last marginal glyph.

With four glyphs allowed in a larger circle rather than three, the effects of marginal glyphs become larger.
- 294 permutations is instead 2,058, and the first marginal glyph adds 1,525 effects to consider.
- 180 permutations without duplication is instead 720, and the first marginal glyph adds 749 new effects to consider.
- 68 combinations disregarding order is instead 124, and the first marginal glyph adds 71 new effects to consider.
- 35 combinations disregarding order and duplicates is still only 35, but the first marginal glyph adds 34 new effects to consider rather than 20.

Weighing the Outcomes
- With any variety of effect-glyph-interaction, there are a large number of effects that would need to be considered
- If there are a large number of interaction effects, Arcanists would have a considerable number of additional options not available to non-Arcanists
- If there is a small number of interaction effects, most permutations or combinations wouldn't produce a unique effect but it would take more time for Arcanists to find the unique effects
- Opening up arcana (with most glyphs available to all, some available only to scholars, and medium circles available to non-Arcanist scholars) is still viable and leaves plenty for Arcanists to research as well as potential grounds for collaboration
- Going with maximum permutations (duplication is distinct, order is important, 4+ glyphs in larger circles), the 6 released glyphs would require at least 50 hours to research, with an additional 40 hours of research with the release of a seventh glyph (and 140k + 112k riln sinks in chalk respectively)
- That would provide a lot of hours of something to do for Arcanists, though in my pessimism I suspect many combinations would eventually be of the 'worst kept secret' variety (think red brass), and those hours would probably be in safer locales rather than out of town/in the wilds exploring
- Modifier glyphs (duration, effect, area, etc.) could also be added to the mix, and the modifier glyphs would naturally be limited based on the number of effect glyphs required to produce the effect

Other paths
The alternative of one effect glyph per circle (or multiple effect glyphs with no effect glyph interaction) and some modifier glyphs would still provide Arcanists with a larger set of options while requiring considerably less contemplation of glyph effects/interactions. Opening up arcana still leaves Arcanists with distinct options if they have access to larger circles (even if they don't have exclusive access to any glyphs other than the circle/shaping glyphs), but with less possibility for collaboration until other avenues of research are offered.
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