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## Filters

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Chromatic polynomials and representations of the symmetric group

2002
*
Linear Algebra and its Applications
*

In

doi:10.1016/s0024-3795(01)00610-3
fatcat:kgo42ulp6ncpzkloxdlsnchsdm
*the*present paper*the*levels are explained by using a version of*the*sieve principle,*and*it is shown that*the*terms at level correspond to*the*irreducible representations of*the**symmetric**group*Sym ... Previous calculations for b = 2*and*b = 3 suggest that*the**chromatic**polynomial*contains terms that occur in 'levels'. ...*symmetric**group*Sym . ...##
###
Galois groups of chromatic polynomials

2012
*
LMS Journal of Computation and Mathematics
*

Most of these

doi:10.1112/s1461157012001052
fatcat:zvgosqoaqrbhndtkipmx2n6zq4
*chromatic**polynomials*have*symmetric*Galois*groups*. ... We give a summary of*the*Galois*groups*of all*chromatic**polynomials*of strongly non-clique-separable graphs of order at most 10*and*all*chromatic**polynomials*of non-clique-separableθ-graphs of order at ... I thank Graham Farr for suggesting this topic*and*for giving regular feedback on drafts of this paper. I thank John Cannon for his support*and*for providing us with a copy of Magma. ...##
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Algebraic properties of chromatic roots
[article]

2016
*
arXiv
*
pre-print

We also report computational results on

arXiv:1610.00424v1
fatcat:zopghzaazzdjpks5m2fzfdgkom
*the*Galois*groups*of irreducible factors of*the**chromatic**polynomial*for some special graphs. Finally, extensions to*the*Tutte*polynomial*are mentioned briefly. ...*The*idea is to consider certain special classes of graphs for which*the**chromatic**polynomial*is a product of linear factors*and*one "interesting" factor of larger degree. ... We are grateful to*the*Institute for*the*excellent facilities*and*opportunities for interaction which it provided. ...##
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Graph coloring-related properties of (generating functions of) Hodge-Deligne polynomials
[article]

2022
*
arXiv
*
pre-print

Finally, we give an application of methods used to symmetries of Hodge-Deligne

arXiv:2203.11930v1
fatcat:ppcuzq32onfsdfrjywbbxy73ve
*polynomials*of varieties*and*their configuration spaces*and*their relation to*chromatic**symmetric**polynomials*. ... Motivated by a connection between*the*topology of (generalized) configuration spaces*and**chromatic**polynomials*, we show that generating functions of Hodge-Deligne*polynomials*of quasiprojective varieties ... ACKNOWLEDGEMENTS I am very thankful to my advisor Benson Farb for helpful discussions*and*encouragement throughout*the*project. ...##
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A categorification of the chromatic symmetric polynomial

2015
*
Discrete Mathematics & Theoretical Computer Science
*

International audience

doi:10.46298/dmtcs.2527
fatcat:ubn3lesmzreahi7oqgctojwj6e
*The*Stanley*chromatic**polynomial*of a graph $G$ is a*symmetric*function generalization of*the**chromatic**polynomial*,*and*has interesting combinatorial properties. ... We also obtain analogues of several familiar properties of*the**chromatic**symmetric**polynomials*in terms of homology. ... Acknowledgements*The*former author would like to thank*the*Simons Foundation for its support via*the*AMS Travel*and*Simons Collaboration grants. ...##
###
A categorification of the chromatic symmetric function

2018
*
Journal of combinatorial theory. Series A
*

*The*Stanley

*chromatic*

*symmetric*

*polynomial*of a graph G is a

*symmetric*function generalization of

*the*

*chromatic*

*polynomial*,

*and*has interesting combinatorial properties. ... We also obtain analogues of several familiar properties of

*the*

*chromatic*

*symmetric*

*polynomials*in terms of homology. Résumé. ... Acknowledgements

*The*former author would like to thank

*the*Simons Foundation for its support via

*the*AMS Travel

*and*Simons Collaboration grants. ...

##
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Chromatic symmetric function of graphs from Borcherds algebras
[article]

2021
*
arXiv
*
pre-print

*The*absolute value of

*the*linear coefficient of

*the*

*chromatic*

*polynomial*of G is known as

*the*

*chromatic*discriminant of G. ... We prove that

*the*

*chromatic*

*symmetric*function of G can be recovered from

*the*Weyl denominator identity of 𝔤

*and*this gives a Lie theoretic proof of Stanley's expression for

*chromatic*

*symmetric*function ... Since,

*the*

*chromatic*

*symmetric*functions are generalization of

*chromatic*

*polynomials*it is natural ask for

*the*connection between

*chromatic*

*symmetric*functions

*and*Borcherds algebras. ...

##
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Algebraic invariants arising from the chromatic polynomials of theta graphs

2014
*
The Australasian Journal of Combinatorics
*

We give a complete description of

dblp:journals/ajc/DelbourgoM14
fatcat:27nx7r7h7jg6hiddiockdtowpq
*the*Galois*group*, discriminant*and*ramification indices for*the**chromatic**polynomials*of theta graphs with three consecutive path lengths. ... This paper investigates some algebraic properties of*the**chromatic**polynomials*of theta graphs, i.e. graphs which have three internally disjoint paths sharing*the*same two distinct end vertices. ... They also thank*the*Australian Research Council for their support under an ARC-DP110100957 grant. ...##
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Coloring With a Limited Paintbox

2022
*
Notices of the American Mathematical Society
*

If you retort with some very simple case which makes me out a stupid animal, I think I must do as

doi:10.1090/noti2490
fatcat:ia2ybtyhlzhxzazz62426mbzoq
*the*Sphynx did. 1 Hamilton wrote back on ... On October 23, 1852 Francis Guthrie asked his brother Frederick, a student at University College London, England, to ask his professor, Augustus De Morgan, whether four colors sufficed to color*the*countries ... stories made writing*and*researching this article all*the*more*chromatic*. ...##
###
Transplanting Trees: Chromatic Symmetric Function Results through the Group Algebra of S_n
[article]

2022
*
arXiv
*
pre-print

One of

arXiv:2112.09937v2
fatcat:hlnlx3k4gzg6bm2grtv7ulq7cq
*the*major outstanding conjectures in*the*study of*chromatic**symmetric*functions (CSF's) states that trees are uniquely determined by their CSF's. ... Additionally, we prove that a "parent function" of*the*CSF defined in*the**group*ring of S_n can uniquely determine trees, providing further support for Stanley's conjecture. ...*The*author Angèle Foley was supported by an NSERC Discovery Grant*and**the*authors Joshua Kazdan*and*Sofía Martínez Alberga are supported by*the*NSF GRFP: Award # 1650114. ...##
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From poset topology to 𝑞-Eulerian polynomials to Stanley's chromatic symmetric functions
[chapter]

2016
*
The Mathematical Legacy of Richard P. Stanley
*

In recent years we have worked on a project involving poset topology, various analogues of Eulerian

doi:10.1090//mbk/100/18
fatcat:gg5lpzddhzcxhcvmptke7eqhdy
*polynomials*,*and*a refinement of Richard Stanley's*chromatic**symmetric*function. ... Here we discuss how Stanley's ideas*and*results have influenced*and*inspired our own work. ... as*the**chromatic**symmetric*function of G n . ...##
###
From Poset Topology to q-Eulerian Polynomials to Stanley's Chromatic Symmetric Functions
[article]

2015
*
arXiv
*
pre-print

In recent years we have worked on a project involving poset topology, various analogues of Eulerian

arXiv:1505.03530v1
fatcat:23qf77nybvenvhg6mawenjhdda
*polynomials*,*and*a refinement of Richard Stanley's*chromatic**symmetric*function. ... Here we discuss how Stanley's ideas*and*results have influenced*and*inspired our own work. ... Since*the*hard Lefschetz map on H * (H(m)) commutes with*the*action of*the**symmetric**group*,*the*conjecture implies that X inc(P (m)) (x; t) is Schur-unimodal, which in turn implies that Conjecture 4.7 ...##
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Orbital Chromatic and Flow Roots

2006
*
Combinatorics, probability & computing
*

*The*

*chromatic*

*polynomial*P Γ (x) of a graph Γ is a

*polynomial*whose value at

*the*positive integer k is

*the*number of proper kcolourings of Γ. ... Our hypotheses include parity conditions on

*the*elements of G

*and*also some special types of graphs

*and*

*groups*. We also look at orbital flow roots. ... Let Γ be

*the*null graph on n vertices,

*and*G

*the*

*symmetric*

*group*S n . ...

##
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The Chromatic Polynomial of a Digraph
[article]

2022
*
arXiv
*
pre-print

This decomposition will confirm

arXiv:1911.09547v3
fatcat:jhmi3sxkjrhqtcgr6mtufluvx4
*the*equality of our*chromatic**polynomial*of a digraph*and**the**chromatic**polynomial*of*the*underlying undirected graph in*the*case of*symmetric*digraphs. ... Furthermore we will decompose our NL-coflow*polynomial*, which becomes*the**chromatic**polynomial*of a digraph by multiplication with*the*number of colors to*the*number of components, examining*the*special ... Probably,*the**chromatic**polynomial*of a graph is better known than*the*flow*polynomial*. ...##
###
Combinatorics of multivariate chromatic polynomials for rooted graphs
[article]

2022
*
arXiv
*
pre-print

Richard Stanley defined

arXiv:2206.05392v2
fatcat:eh4i5u76i5h4tf52yjosp3576a
*the**chromatic**symmetric*function X_G of a graph G*and*conjectured that trees T*and*U are isomorphic if*and*only if X_T=X_U. ...*The*first is that our*polynomials*satisfy*the*analogue of Stanley's conjecture: two rooted trees are isomorphic as rooted graphs if*and*only if their rooted*chromatic**polynomials*are equal. ... Acknowledgments*The*authors thank Sergei Chmutov, Peter McNamara, Rosa Orellana, John Shareshian, Stephanie van Willigenburg,*and*Victor Wang for supplying references*and*other helpful comments regarding ...
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