A Parameterized Formulation for the Maximum Number of Runs Problem.
Bibliographic record
Abstract
Abstract. A parameterized approach to the problem of the maximum number of runs in a string was introduced by Deza and Franek. In the approach referred to as the d-step approach, in addition to the usual parameter the length of the string, the size of the string’s alphabet is considered. The behaviour of the function ρd(n), the maximum number of runs over all strings of length n with exactly d distinct symbols, can be handily expressed in the terms of properties of a table referred to as the (d, n−d) table in which ρd(n) is the entry at the dth row and (n−d)th column. The approach leads to a conjectured upper bound ρd(n) ≤ n − d for 2 ≤ d ≤ n. The parameterized formulation shows that the maximum within any column of the (d, n − d) table is achieved on the main diagonal, i.e. for n = 2d, and motivates the investigation of the structural properties of the run-maximal strings of length n bounded by a constant times the size of the alphabet d. We show that ρd(n) = ρn−d(2n − 2d) for 2 ≤ d ≤ n < 2d, ρd(2d) ≤ ρd−1(2d − 1) + 1 for d ≥ 3, ρd−1(2d − 1) = ρd−2(2d − 2) = ρd−3(2d − 3) for d ≥ 5, and {ρd(n) ≤ n − d for 2 ≤ d ≤ n} ⇔ {ρd(9d) ≤ 8d for d ≥ 2}. The results allow for an efficient computational verification of entries in the (d, n − d) table for higher values of n and point to a plausible way of either proving the maximum number of runs conjecture by showing that possible counter-examples on the main diagonal would exhibit an impossible structure, or to discover an unexpected counter-example on the main diagonal of the (d, n−d) table. This approach provides a purely analytical proof of ρd(2d) = d for d ≤ 15 and, using the computational results of ρ2(d + 2) for d = 16,..., 23, a proof of ρd(2d) = d for d ≤ 23.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.022 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.005 | 0.012 |
| Open science | 0.005 | 0.003 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.015 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".