C1/C2 common collation  is a method of further splitting up the two sheets used in sets with 101 commons and 11 × 11 sheets. This method is used for most US-printed sets with 101 commons. (Most large sets since Shards of Alara have 101 commons, at least when counting for this purpose.)
Each sheet is split up into a run of 66 cards and a run of 55 cards. (I don't have any direct evidence of how this split related to the sheet, but given lots of prior examples of splitting sheets into the top 6 rows and bottom 5 rows, that seems likely here as well.) From the 60-card sheet, this gives a run with 33 cards each appearing twice, called A, and a run with 27 cards each appearing twice plus the short-printed card, called C1. The other sheet gives a run with 22 cards each appearing three times, called B, and a run with 18 cards each appearing three times plus the short-printed card, called C2.
Each pack will contain cards from A followed by cards from B followed by either cards from C1 or C2. (In packs where the rare in is front, the order is reversed, although the Japanese printings most known for rare-in-front collation don't use C1/C2.)
In more recent C1/C2 sets, the cards in the A run are often mono-colored cards of three particular colors (the exact choice of which may vary from set to set, but usually Jeskai or Mardu) in an alternating pattern. The cards in the B run are then alternating mono-colored cards of the other two colors. This helps ensure that each pack is likely to have a mono-colored common of every color (although this condition still may not be guaranteed, especially accounting for foils).
The first generation consists of sets between Shards of Alara and Innistrad. (Innistrad doesn't fall cleanly into any category because of double-faced cards, and Shards of Alara itself is probably nonstandard because it has an elevated foil rate.) This is the generation that I have the I understand the least, as there are many pack types, the foil collation is complicated, and data is more limited.
Exactly half of packs are C1 and half are C2. Not counting foils, C1 packs always have 3 A cards, 1-2 B cards, and 5-6 C cards. 2B packs are more common, but the exact rate is not known. (This rate will be further discussed in relation to foils.) For C2 packs, not counting foils, packs can have either 3 or 4 A cards, each with equal probability. 3A packs have 1-4 B cards and 4-6 C cards. The rates of these pack types known, but 3A packs with only one B card stand out as being rare. 4B packs have 3-4 B cards and 2-3 C cards. 3B packs seem to occur 2/3 of the time, though this data is far from certain.
Foils can mostly be understood as appearing in certain packs types displacing a common from a certain run. Foil uncommons displace A commons. Probably they are equally likely in all pack types. Foil rares displace C1 commons (and thus appear only in C1 packs). Foil commons that appear in C1 packs simply displace a B common (which can sometimes result in a pack with no B commons). Foil commons that appear in C2 packs are more complicated, though they are still related to what the contents of the pack would have been if a foil hadn't appeared. (The hypothetical contents of the pack if it had no foil are clear when looking at the sequential pattern of pack types within a booster box.) In the 3A case, packs that would have 1-2 B cards now get 1 B card, and packs with with 3-4 B cards get 2 B cards. In the 4A case, there is always 1 B card. Foil commons are much more likely to appear in C2 packs, but are probably equally likely to occur in each type of C2 pack, and, given they appear in a C1 pack, probably equally likely to occur in each kind of pack of that type.
Except for Shards of Alara, these sets have an advertised foil odds of aproximately 1 : 67 cards which I interpret as 15/67 packs. I am pretty confident from collation patterns that 1/8 of this comes from foil commons in C2 packs. (So, given that foils also exist in C1 packs, the overall common foil rate is greater than 1/8.) The other numbers are less certain, but the foil uncommon rate is probably greater than 1/20 and the foil rare rate is probably greater than 1/32.
Application of sheet math on the B sheet is not very useful because the unknown rates of the C2 pack types can alter the balance of the B and C2 runs. For the A sheet, the foil rates for uncommons and rares balance the A and C1 runs, but this also depends on the unknown rate of C1 2B packs. The inter-sheet 3 : 2 constraint can related the rate of C1 2B packs the the foil common rate (as equivalent to the sum of the foil uncommon and rare rates given the total rate).
Also, the total as-fan of A cards is known up to the foil uncommon rate. This allows using the constraints to compute the foil uncommon rate by comparing against the total common as-fan (which is based on the total foil rate). This gives a number about 1% greater than 1/20. This is close to what I expect from collation patterns, but empirically, I think 1% is not enough of a difference over 1/20. (This could be accounted for by inexactitude of 15/67 or of one of the constraints, or even by unobserved outliers of the common model.)
Similarly considering the as-fan of C1 cards, we can relate the foil rare rate to the rate of 2B C1 packs. A foil rare rate of 1/32 would correspond to about 3/5 of C1 packs being 2B, so this is probably reasonable as a crude estimate of that rate.
Avacyn Restored doesn't fall cleanly into the first generation or the second generation. It has the simplified C2 collation of second generation sets, but it doesn't have 2A packs as far as I've seen. Further investigation may be required.
Return to Ravnica marks the start of the second generation. This type of collation has some 2A packs, and the collation of C2 packs and foils is somewhat simplified.
Again, exactly half of packs are C1 and half are C2. Not counting foils, C1 packs have 2-3 A cards, 2 B cards, and 4-5 C cards. 3A packs appear 2/3 of the time. For C2 packs, again not counting foils, packs have 4 A cards, 2-4 B cards, and 2-4 C cards. Some sets may not have 4+3+3 packs. Other authors' write-ups of early second generation sets don't mention this pack type. The earliest I've confirmed their appearance is Khans of Tarkir, but it is sometimes hard to prove they don't exist as this is the rarest pack type even when it is possible. (The introduction of this third C2 pack type shouldn't be needed to meet any sheet math constraints given full control over the rate of 2B versus 4B packs, but maybe it simplifies some practical considerations.) For sets where all of 2B, 3B, and 4B appear, probably the rate of 2B packs is 4/5. The rate between 3B and 4B is not known, but 4B is probably more common.
Starting with Khans of Tarkir, sets tend to assign three colors (usually red, white, and blue; or, less commonly, red, white, and black) to the A run and the other two colors to the B run. This can help increase the chance of getting a common of each color in every booster pack, although this is still uncertain because of 2A packs and foils.
Foils can be understood as appearing in certain pack types displacing a common from a certain run. Foil commons displace A cards, foil uncommons displace B cards, and foil rares displace C2 cards. (For sets that have an additional foil sheet, those foils will often displace C1 cards.)
The foil rates of each rarity are not completely fixed throughout this generation, and these rates are related to the kinds of packs where each rarity of foil can appear. The common foil rate is always close to 1/8, but it can be greater or less than that number depending on the set. For Return to Ravnica and Gatecrash, there are indications of a greater rate. In Theros, I detected no deviation (although a rare deviation could have gone unobserved). In Khans of Tarkir and Dragons of Tarkir, there are indications of a lesser rate. Subsequent C1/C2 sets had special foils (like masterpieces or double-faced cards) until Dominaria which I'm pretty sure is exactly 1/8. (My data set for Dominaria is large with no exceptions for this number, but it is always impossible to rule out rare outliers.) Before Dominaria, foil commons would appear mostly or all in C1 packs. For sets with common foil rate less than or equal to 1/8, commons would appear all in C1 packs. Otherwise, only those foil cards accounting for the difference from the 1/8 rate would appear in C2 packs. Starting with Dominaria, common foils were switched to C2 packs. (Displacing A commons from C2 packs is preferable given the color split between the A and B runs.) The 1/8 rate seems to be standard for the remainder of the generation.
In all second generation sets, I think the foil uncommon rate is greater than 1/20 (though it's possible the exact rate varies). Foil uncommons are probably equally likely to appear in C1 and C2 packs. In C1 packs, they are probably indepedent of pack type, but in C2 packs, they probably always appear in 2B packs.
For foil rares, there is probably no preference for pack type besides appearing only in C2 packs. (It is hard speak of 4B and especially 3B packs due to their rarity, but at least in Dominaria, foil rares have been observed in all C2 pack types.)
The A sheet constraint is powerful for second generation sets. Taking the common model as given, it allows directly calculation of the common foil rate as 2/15, slightly higher than 1/8. This is plausible as the actual exact foil rate for those early second generation sets where the pattern suggests a rate greater than 1/8. But, it also suggests that this constraint might be violated otherwise (meaning A commons are more likely than C1 commons, and a small fraction of printed cards are wasted). This effect would be mitigated in sets that have an extra foil sheet that displaces C1. Applying the B sheet math constraint is not very effective as it balances the uncommon and rare foil rates with the unknown split between B and C2 cards in C2 packs.
The inter-sheet 3 : 2 constraint is also interesting. The common model fully specifies the as-fan of each sheet (as the split between B and C2 doesn't matter here) except for foils, and as uncommon and rare foils displace the same sheet, they can be considered together and related to the common foil rate via the total rate. Thus, if we take a common foil rate as given, we can calculate the total rate given this constraint. For 1/8 common rate, the calculated total rate is 5/24 which is significantly too low (1/72 cards). This could imply that the A sheet is slightly overprinted in sets that use this 1/8 (or less). For 2/15 common rate, this constraint implies a 2/9 total rate (1/67.5 cards) which actually matches the advertised rate up to rounding.
The third generation starts with the increase of the foil rate to 1/3 packs in Core Set 2020. It is also characterized by the introduction of showcase cards and other new special treatments which followed in the next set.
Pack structure is mostly the same as the second generation. Ignoring foils, the types of packs that appear are the same as those from the second generation at the same rates. 3B packs appear in these sets, but their rate is still unknown (and may be different than it is for second generation sets and may depend on how foils work for each set). For sets where showcase commons appear in boosters, the showcase commons are on the B sheet (in run B or run C2) where each card appears three times. The showcase version replaces one of the three normal versions of the card.
Core Set 2020, the only third generation set without showcase cards, needs to be considered separately (but note that it has a smaller sample size so I am less confident in any results). The common and uncommon foil rates were observed to be 1/5 and 1/12, respectively (making the rare foil rate 1/20). The elevated common foil rate (compared to the second generation) means that common foils now appear in both C1 and C2 packs (at equal rates). However, there is probably no possibility for a 1A pack because common foils will replace C1 cards from 2A packs, and still replace an A card otherwise. There is probably no other connection with C1 pack type, but it could be that common foils are biased against 2B C2 packs (enough to bring their rate up to 1/4 in 3B and 4B packs). As usual, uncommon foils and rare foils replace B and C2 cards, respectively. For uncommon foils, probably half occur in C2 packs (no correlation with type) and half occur in 2A C1 packs. Rare foils probably have no pack type correlation (except, of course, appearing in C2 packs).
Applying the A sheet constraint to this model again gives leverage on the common foil rate. As in the second generation, if all common foils displace A cards, the foil rate should be 2/15 to balance the constraint. 1/5 is significantly higher than that, so having some common foils displace C1 cards instead helps balance things out. However, under the model I've suggested, C1 cards are replaced only in 1/5 of 2A packs which leaves A cards as still being slightly too common relative to C1 cards (although this effect is even smaller than the unbalance from 1/8 common foil rate in the second generation). (It could also be that very rare 1A packs exist.)
The inter-sheet constraint is also directly applicable with just the common and total foil rates. This constraint is perfectly balanced with 1/5 common foils and 1/3 total.
The B sheet constraint again involves the rate of 3B packs, so the predictions for the uncommon and rare foil rates can be applied to calculate this rate. This suggests 5/66 of non-2B C2 packs are 3B. (Note, I didn't actually confirm a sighting of a 3B pack for this set, but this is not unusual given the sample size.) (It's also possible that the rate is not exactly 5/66 and the sheet is unbalanced.)
Showcase foils complicate things. In sets with showcase foils, I've observed a 1/6 common foil rate (lower than Core Set 2020) and a 1/12 uncommon foil rate (the same as Core Set 2020). I don't have indepedent predictions for the rare or showcase rates based on the data, but we have data directly from Wizards that gives an approximate 1/34 showcase (plus borderless) foil rate from Eldraine (which makes the rare rate about 11/204). It is unknown if this value holds for others sets (which, after all, have different numbers of showcase cards). From Core Set 2021, Wizards gave the approximate number 1/35 for foil showcase and borderless cards, but the (non-foil non-planeswalker) borderless cards in that set make it an unusual example.
In these sets, common foils, uncommon foils, rare foils, and showcase (plus borderless planeswalker) foils displace A, B, C2, and C1 commons respectively. Common foils no longer displace C1 cards; they displace A cards exclusively. Common foils can appear in all packs types. They probably have no bias in C2 packs, but in C1 packs, they are biased toward 2A packs with probability boosted to 1/4 in packs of that type. Uncommon foils should have the same pack type correlation as Core Set 2020.
The A sheet constraint balances the common foil rate against the showcase foil rate. Using 1/5 and 1/34 for those rates, there is an extremely small bias in favor of A. (The possible uncertainty error in 1/34 versus, say, 1/35 wouldn't account for the full discrepancy, though.)
Again, the inter-sheet constraint can be applied with these two foil rates plus the total foil rate to show an extremely small bias in favor of sheet A.
And again, the B sheet constraint can be used to estimate the rate of 3B packs using the uncommon and rare foil rates. Using 11/204 as the rare foil rate gives that 109/1122 of non-2B C2 packs are 3B. (I haven't observed 3B packs for Eldraine, but I have seen them for other third generation sets.)
There is another style of C1/C2 collation that has been observed concurrently with second generation collation, although it has so far only been observed for two sets, Core Set 2021 and Modern Horizons. The most obvious difference for this kind of printing is that the cards are in reverse order: packs have rares in the front, but also commons in C, B, A order. The sheet layouts match the sheet layouts for the classic US printings. The details of these printings are discussed further on the pages for those sets.
 This method of collation was discovered by user Medussa on the MTG Salvation forums.