Betti diagrams from graphs

Engström, Alexander; Stamps, Matthew T.

doi:10.2140/ant.2013.7.1725

Cited by 8 publications

(17 citation statements)

References 17 publications

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“…Following the discussion around converting Betti diagrams and Boij-Söderberg coefficients into each other from the paper [9], we record these pure diagrams as the two vectors…”

Section: Boij-söderberg Theorymentioning

confidence: 99%

“…To a 2-linear ideal (equivalently, a graph with chordal complement) we can associate a unique anti-lecture hall composition with t = 1 and λ 1 = 1, see Section 4 of [9]. 3 The Boij-Söderberg theory of ideals of Booth-Lueker graphs…”

Section: Anti-lecture Hall Compositionsmentioning

confidence: 99%

“…Following the discussion around converting Betti diagrams and Boij-Söderberg coefficients into each other from the paper [9], we record these pure diagrams as the two vectors The zeros in the end extend indefinitely, and it will be convenient for us to write a specific amount of them, as it will become clear in the next sections. We also denote from now on c s := c πs .…”

Section: Boij-söderberg Theorymentioning

confidence: 99%

“…Engström and Stamps showed in [9] how Betti tables and Boij-Söderberg coefficients are related for 2-linear resolutions. We summarise this in the following lemma.…”

Section: On 2-linear Resolutionsmentioning

confidence: 99%

“…It turns out that the information carried over to the algebraic setting from the graphs are all compiled in its degree vector. We will employ several results by Engström and Stamps [9] on 2-linear resolutions and Boij-Söderberg theory to reach these goals.…”

Section: On 2-linear Resolutionsmentioning

confidence: 99%

See 4 more Smart Citations

Explicit Boij–Söderberg theory of ideals from a graph isomorphism reduction

Engström

Jakobsson

Orlich

2020

Journal of Pure and Applied Algebra

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In the origins of complexity theory Booth and Lueker showed that the question of whether two graphs are isomorphic or not can be reduced to the special case of chordal graphs. To prove that, they defined a transformation from graphs G to chordal graphs BL(G). The projective resolutions of the associated edge ideals I BL(G) is manageable and we investigate to what extent their Betti tables also tell non-isomorphic graphs apart. It turns out that the coefficients describing the decompositions of Betti tables into pure diagrams in Boij-Söderberg theory are much more explicit than the Betti tables themselves, and they are expressed in terms of classical statistics of the graph

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“…Following the discussion around converting Betti diagrams and Boij-Söderberg coefficients into each other from the paper [9], we record these pure diagrams as the two vectors…”

Section: Boij-söderberg Theorymentioning

confidence: 99%

Section: Anti-lecture Hall Compositionsmentioning

confidence: 99%

Section: Boij-söderberg Theorymentioning

confidence: 99%

“…Engström and Stamps showed in [9] how Betti tables and Boij-Söderberg coefficients are related for 2-linear resolutions. We summarise this in the following lemma.…”

Section: On 2-linear Resolutionsmentioning

confidence: 99%

Section: On 2-linear Resolutionsmentioning

confidence: 99%

See 3 more Smart Citations

Explicit Boij–Söderberg theory of ideals from a graph isomorphism reduction

Engström

Jakobsson

Orlich

2020

Journal of Pure and Applied Algebra

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Betti numbers of fat forests and their Alexander dual

Fröberg

2022

J Algebr Comb

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Let k be a field and $$R=k[x_1,\ldots ,x_n]/I=S/I$$ R = k [ x 1 , … , x n ] / I = S / I a graded ring. Then R has a t-linear resolution if I is generated by homogeneous elements of degree t, and all higher syzygies are linear. Thus, R has a t-linear resolution if $$\mathrm{Tor}^S_{i,j}(S/I,k)=0$$ Tor i , j S ( S / I , k ) = 0 if $$j\ne i+t-1$$ j ≠ i + t - 1 . For a graph G on $$\{1,\ldots ,n\}$$ { 1 , … , n } , the edge algebra is $$k[x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I , where I is generated by those $$x_ix_j$$ x i x j for which $$\{ i,j\}$$ { i , j } is an edge in G. We want to determine the Betti numbers of edge rings with 2-linear resolution. But we want to do that by looking at the edge ring as a Stanley–Reisner ring. For a simplicial complex $$\Delta $$ Δ on $$[\mathbf{n}]=\{1,\ldots ,n\}$$ [ n ] = { 1 , … , n } and a field k, the Stanley–Reisner ring $$k[\Delta ]$$ k [ Δ ] is $$k[x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I , where I is generated by the squarefree monomials $$x_{i_1}\ldots x_{i_k}$$ x i 1 … x i k for which $$\{ i_1,\ldots ,i_k\}$$ { i 1 , … , i k } does not belong to $$\Delta $$ Δ . Which Stanley–Reisner rings that are edge rings with 2-linear resolution are known. Their associated complexes has had different names in the literature. We call them fat forests here. We determine the Betti numbers of many fat forests and compare our result with what is known. We also consider Betti numbers of Alexander duals of fat forests.

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The mathematics of lecture hall partitions

Savage

2016

Journal of Combinatorial Theory, Series A

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Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to lecture hall partitions have used polyhedral geometry to discover further properties of these rich combinatorial objects.In this paper we give an overview of some of the surprising connections that have surfaced in the process of trying to understand the lecture hall partitions.

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Betti diagrams from graphs

Cited by 8 publications

References 17 publications

Explicit Boij–Söderberg theory of ideals from a graph isomorphism reduction

Explicit Boij–Söderberg theory of ideals from a graph isomorphism reduction

Betti numbers of fat forests and their Alexander dual

The mathematics of lecture hall partitions

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