Interleaving: Difference between revisions
m →Lemma |
|||
| Line 37: | Line 37: | ||
If ''S'' consists of the subwords '''XZ''' and '''YZ''' arranged in the pattern of a single-period binary circular word ''w''(''x'', ''y'') where |''w''| > 2, and ''k'' is an odd number greater than 1 and less than |''S''| - 1, then the class of ''k''-steps has more than 3 abstract intervals. | If ''S'' consists of the subwords '''XZ''' and '''YZ''' arranged in the pattern of a single-period binary circular word ''w''(''x'', ''y'') where |''w''| > 2, and ''k'' is an odd number greater than 1 and less than |''S''| - 1, then the class of ''k''-steps has more than 3 abstract intervals. | ||
{{proof| Denote by |''w''| the length of subword ''w'' in letters and by ‖''w''‖ the interval subtended by subword ''w'' in its circular word. | |||
Let ''A'' be the set of all (''k'' - 1)/2-step intervals of ''w''. |''A''| = 1 implies that ''k'' = 1 mod |''w''|, so |''A''| ≥ 2 and contains at least two intervals '''w'''<sub>1</sub> and '''w'''<sub>2</sub>. | Let ''A'' be the set of all (''k'' - 1)/2-step intervals of ''w''. |''A''| = 1 implies that ''k'' = 1 mod |''w''|, so |''A''| ≥ 2 and contains at least two intervals '''w'''<sub>1</sub> and '''w'''<sub>2</sub>. | ||
| Line 80: | Line 80: | ||
(c) is similar to (b). | (c) is similar to (b). | ||
}} | |||
=== Attempted proof of Conjecture === | === Attempted proof of Conjecture === | ||