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

Wang, David G. L.; Wang, Monica M. Y.

doi:10.1007/s10801-022-01175-6

Cited by 2 publications

(1 citation statement)

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“…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

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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

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