On the $e$-Positivity of $(claw, 2K_2)$-Free Graphs

Li, Grace M.X.; Yang, Arthur L. B.

doi:10.37236/9910

Cited by 5 publications

(2 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Wolfgang III [28] establishes a criterion for the e-positivity in terms of connected partitions of G. Dahlberg, Foley, and van Willigenburg [4] presented a family of claw-free graphs that are neither e-positive nor contractible to the claw. Some special graphs are known to be e-positive, see Cho and Huh [3], Foley, Hoàng, and Merkel [6], Gebhard and Sagan [8], Li, Li, Wang, and Yang [11], Li and Yang [12], Wang and Wang [27] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative Symmetric Functions, Lie Series and Descent Algebras

Thibon¹

2021

Algebra and Applications 2

View full text Add to dashboard Cite

We obtain the Schur positivity of spider graphs of the forms S(a, 2, 1) and S(a, 4, 1), which are considered to have the simpliest structures for which the Schur positivity was unknown. The proof outline has four steps. First, we find noncommutative analogs for the chromatic symmetric functions of the spider graphs S(a, b, 1). Secondly, we expand the analogs under the Λ-and R-bases, whose commutative images are the elementary and skew Schur symmetric functions, respectively. Thirdly, we recognize the Schur coefficients via the Littlewood-Richardson rule in terms of norms of multisets of Yamanouchi words. At last we establish the Schur positivity combinatorially together with the aid of computer assistance. Contents 1. Introduction 2. Preliminaries 2.1. Compositions, partitions and tableaux 2.2. Ribbons and ribbon tableaux 2.3. Commutative symmetric functions 2.4. Noncommutative symmetric functions 2.5. Maps between multisets 3. Noncommutative analogs of X Pn and X S(a,b,1) 4. The spiders S(a, 2, 1) and S(a, 4, 1) are Schur positive 4.1. The spiders S(a, 2, 1) are Schur positive 4.2. The spiders S(a, 4, 1) are Schur positive References Appendix A. A proof of Proposition 4.5

show abstract

Section: Introductionmentioning

confidence: 99%

Noncommutative Symmetric Functions, Lie Series and Descent Algebras

Thibon¹

2021

Algebra and Applications 2

View full text Add to dashboard Cite

show abstract

“…The incomparability graph of P is the graph obtained by taking the elements of P as its vertices and letting two vertices be adjacent if and only if they are not comparable in P . Stanley's (3 + 1)-free conjecture has received considerable attention, see for instance [1,3,5,7,8,9,10,12,13,17,18] and references therein.…”

Section: Introductionmentioning

confidence: 99%

The twinning operation on graphs does not always preserve $e$-positivity

H.¹,

Li²,

Wang³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

Motivated by Stanley's (3 + 1)-free conjecture on chromatic symmetric functions, Foley, Hoàng and Merkel introduced the concept of strong e-positivity and conjectured that a graph is strongly e-positive if and only if it is (claw, net)-free. In order to study strongly e-positive graphs, they further introduced the twinning operation on a graph G with respect to a vertex v, which adds a vertex v ′ to G such that v and v ′ are adjacent and any other vertex is adjacent to both of them or neither of them. Foley, Hoàng and Merkel conjectured that if G is e-positive, then so is the resulting twin graph G v for any vertex v. Based on the theory of chromatic symmetric functions in non-commuting variables developed by Gebhard and Sagan, we establish the e-positivity of a class of graphs called tadpole graphs. By considering the twinning operation on a subclass of these graphs with respect to certain vertices we disprove the latter conjecture of Foley, Hoàng and Merkel. We further show that if G is e-positive, the twin graph G v and more generally the clan graphs G (k) v (k ≥ 1) may not even be s-positive, where G (k) v is obtained from G by applying k twinning operations to v.

show abstract

The e-positivity of two classes of cycle-chord graphs

Wang

2022

J Algebr Comb

View full text Add to dashboard Cite

On the $e$-Positivity of $(claw, 2K_2)$-Free Graphs

Cited by 5 publications

References 10 publications

Noncommutative Symmetric Functions, Lie Series and Descent Algebras

Noncommutative Symmetric Functions, Lie Series and Descent Algebras

The twinning operation on graphs does not always preserve $e$-positivity

The e-positivity of two classes of cycle-chord graphs

Contact Info

Product

Resources

About