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The $e$-vector of a simplicial complex

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Wayne A. Johnson, Wiktor J. Mogilski

Combinatorics · Submitted 13 Jun 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We then prove a simple combinatorial identity for the $e$-vector in the case where $\Delta$ is an Eulerian manifold.

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arXiv ID1806.05239 (version 2)
Pith IDH7RB32QH, Pith's permanent record of this paper and its review
CitationWayne A. Johnson, Wiktor J. Mogilski. "The $e$-vector of a simplicial complex." arXiv:1806.05239 (2018). https://pith.science/paper/H7RB32QH
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,072 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.22 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We study the exponential Hilbert series (both coarsely- and finely-graded) of the Stanley-Reisner ring of an abstract simplicial complex, $\Delta$, and we introduce the $e$-vector of $\Delta$, which relates to the coefficients of the exponential Hilbert series. We explore the relationship of the $e$-vector with the classical $f$-vector and $h$-vector of $\Delta$ while simultaneously investigating the geometric information that the $e$-vector encodes about $\Delta$. We then prove a simple combinatorial identity for the $e$-vector in the case where $\Delta$ is an Eulerian manifold.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the exponential Hilbert series (both coarsely- and finely-graded) of the Stanley-Reisner ring of an abstract simplicial complex, $\Delta$, and we introduce the $e$-vector of $\Delta$, which relates to the coefficients of the exponential Hilbert series. contribution 1.00 0.93 0.14 asserted 1.00
s1 We explore the relationship of the $e$-vector with the classical $f$-vector and $h$-vector of $\Delta$ while simultaneously investigating the geometric information that the $e$-vector encodes about $\Delta$. contribution 1.00 0.88 0.13 asserted 0.74
s2 We then prove a simple combinatorial identity for the $e$-vector in the case where $\Delta$ is an Eulerian manifold. result 0.95 0.95 0.29 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
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  • full text
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  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We study the exponential Hilbert series (both coarsely- and finely-graded) of the Stanley-Reisner ring of an abstract simplicial complex, $\Delta$, and we introduce the $e$-vector of $\Delta$, which relates to the coefficients of the exponential Hilbert series. We explore the relationship of the $e$-vector with the classical $f$-vector and $h$-vector of $\Delta$ while simultaneously investigating the geometric information that the $e$-vector encodes about $\Delta$. We then prove a simple combinatorial identity for the $e$-vector in the case where $\Delta$ is an Eulerian manifold.

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