Cyclic sieving on permutations: An analysis of maps and statistics in the FindStat database
Bibliographic record
Abstract
We perform a systematic study of permutation statistics and bijective maps on permutations using SageMath to search the FindStat combinatorial statistics database to identify apparent instances of the cyclic sieving phenomenon (CSP). Cyclic sieving occurs on a set of objects, a statistic, and a map of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when the evaluation of the statistic generating function at the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th power of the primitive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th root of unity equals the number of fixed points under the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th power of the map. Of the apparent instances found in our experiment, we prove 34 new instances of the CSP and conjecture three more. Our results are organized largely by orbit structure, proving instances of the CSP for involutions with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript n minus 1"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">2^{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fixed points and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript left floor StartFraction n Over 2 EndFraction right floor"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="false" stretchy="false"> ⌊ </mml:mo> <mml:mfrac> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo fence="false" stretchy="false"> ⌋ </mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">2^{\lfloor \frac {n}{2}\rfloor }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fixed points, as well as maps whose orbits all have the same size. The FindStat maps which exhibit the CSP include a map constructed by Corteel (using a bijection of Foata and Zeilberger) to swap the number of nestings and crossings, the invert Laguerre heap map, a map of Alexandersson and Kebede designed to preserve right-to-left minima, conjugation by the long cycle, as well as reverse, complement, rotation, Lehmer code rotation, and toric promotion. Our results combined with prior work of the last 5 authors in (Math. Comp. <bold>93</bold> , 921–976) show that, contrary to common expectations, actions that exhibit homomesy are not necessarily the best candidates for the CSP, and vice versa.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot 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.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".