Catalan paths and quasi-symmetric functions
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Bibliographic record
Abstract
We investigate the quotient ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the ring of formal power series <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q left-bracket left-bracket x 1 comma x 2 comma ellipsis right-bracket right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}[[x_1,x_2,\ldots ]]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over the closure of the ideal generated by non-constant quasi-symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and above the line <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y equals x minus k"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">y=x-k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We investigate as well the quotient ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript n"> <mml:semantics> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">R_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of polynomial ring in <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> variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript n"> <mml:semantics> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">R_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded above by the <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 Catalan number.
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 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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| 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 it