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Record W2987306417 · doi:10.1145/3359589

Solving the Sigma-Tau Problem

2019· article· en· W2987306417 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueACM Transactions on Algorithms · 2019
Typearticle
Languageen
FieldEngineering
Topicgraph theory and CDMA systems
Canadian institutionsUniversity of Guelph
Fundersnot available
KeywordsCombinatoricsMathematicsSigmaSuccessor cardinalCayley graphHamiltonian pathPermutation (music)Transposition (logic)Discrete mathematicsGraphTraverseDirected graph

Abstract

fetched live from OpenAlex

Knuth assigned the following open problem a difficulty rating of 48/50 in The Art of Computer Programming Volume 4A : For odd n ≥ 3, can the permutations of { 1,2,… , n } be ordered in a cyclic list so that each permutation is transformed into the next by applying either the operation σ, a rotation to the left, or τ, a transposition of the first two symbols? The Sigma-Tau problem is equivalent to finding a Hamilton cycle in the directed Cayley graph generated by σ = (1 2 ⋅ n ) and τ = (1 2). In this article, we solve the Sigma-Tau problem by providing a simple O ( n )-time successor rule to generate successive permutations of a Hamilton cycle in the aforementioned Cayley graph.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.000
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Simulation or modeling · Consensus signal: none
GenreCandidate signal: Empirical · Consensus signal: none
Teacher disagreement score0.900
Threshold uncertainty score0.646

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0000.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.001

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.

Opus teacher head0.008
GPT teacher head0.197
Teacher spread0.189 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it