Generator-offset property: Difference between revisions
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Λ<sub>2</sub> has a chunk of βs (after the first β’) of size x = either floor(n/μ) (≥ floor(n/floor(n/2)) = 2) or ceil(n/μ) (= floor(n/μ) + 1). Hence Λ<sub>3</sub> has a chunk of γs of that same size. Λ<sub>3</sub> also has a chunk that goes over n and loops back around, which must be of size y = at least 2*(floor(n/μ) − 1) + 1 (Λ<sub>3</sub> might have chunks of size floor(n/μ) − 1 and floor(n/μ) instead) = 2*floor(n/μ) − 1, and at most 2*(floor(n/μ) + 1) + 1 = 2*floor(n/μ) + 3 (if Λ<sub>3</sub> has chunks of size floor(n/μ) and ceil(n/μ)). The difference between the chunk sizes is y − x, which must be 0 or 1, since Λ<sub>3</sub> is a mos. We thus have the following cases: | Λ<sub>2</sub> has a chunk of βs (after the first β’) of size x = either floor(n/μ) (≥ floor(n/floor(n/2)) = 2) or ceil(n/μ) (= floor(n/μ) + 1). Hence Λ<sub>3</sub> has a chunk of γs of that same size. Λ<sub>3</sub> also has a chunk that goes over n and loops back around, which must be of size y = at least 2*(floor(n/μ) − 1) + 1 (Λ<sub>3</sub> might have chunks of size floor(n/μ) − 1 and floor(n/μ) instead) = 2*floor(n/μ) − 1, and at most 2*(floor(n/μ) + 1) + 1 = 2*floor(n/μ) + 3 (if Λ<sub>3</sub> has chunks of size floor(n/μ) and ceil(n/μ)). The difference between the chunk sizes is y − x, which must be 0 or 1, since Λ<sub>3</sub> is a mos. We thus have the following cases: | ||
(In the following, chunk of Λ<sub>2</sub> means chunk of β, and chunk of Λ<sub>3</sub> means chunk of γ.) | |||
====== Case 3.1 ====== | |||
(x, y) = (floor(n/μ), 2*floor(n/μ) − 1) | |||
Since y − x = floor(n/μ) − 1 and floor(n/μ) ≥ 2, we have: x = floor(n/μ) = 2 and y − x = 1; hence y = 2*floor(n/μ) − 1 = 3. The chunk in Λ<sub>3</sub> whose size is 3 is made from two chunks in Λ<sub>2</sub> of size 1. (So Λ<sub>2</sub> has chunks of size 1 and 2, and Λ<sub>3</sub> has chunks of size 2 and 3.) | Since y − x = floor(n/μ) − 1 and floor(n/μ) ≥ 2, we have: x = floor(n/μ) = 2 and y − x = 1; hence y = 2*floor(n/μ) − 1 = 3. The chunk in Λ<sub>3</sub> whose size is 3 is made from two chunks in Λ<sub>2</sub> of size 1. (So Λ<sub>2</sub> has chunks of size 1 and 2, and Λ<sub>3</sub> has chunks of size 2 and 3.) | ||
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(This also implies S is SV3.) | (This also implies S is SV3.) | ||
====== Remaining sub-cases ====== | |||
* Case 3.2: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ) − 1) is impossible because y = 2*floor(n/μ) − 1 can only occur if Λ<sub>3</sub> has chunks of size floor(n/μ) − 1 and floor(n/μ), which contradicts the size of x. | |||
End Case 3.2] | End Case 3.2] | ||
* Case 3.3: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ)) is impossible because y = 2*floor(n/μ) can only occur if Λ<sub>3</sub> has chunks of size floor(n/μ) − 1 and floor(n/μ), which contradicts the size of x. | |||
The remaining cases are all impossible because they imply y − x ≥ 2: | The remaining cases are all impossible because they imply y − x ≥ 2: | ||
* Case 3.4: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ) + 1) | |||
* Case 3.5: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ) + 2) | |||
* Case 3.6: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ) + 3) | |||
* Case 3.7: (x, y) = (floor(n/μ), 2*floor(n/μ)) | |||
* Case 3.8: (x, y) = (floor(n/μ), 2*floor(n/μ) + 1) | |||
* Case 3.9: (x, y) = (floor(n/μ), 2*floor(n/μ) + 2) | |||
* Case 3.10: (x, y) = (floor(n/μ), 2*floor(n/μ) + 3) | |||
== Open conjectures == | == Open conjectures == | ||