Journal of Combinatorics

Volume 9 (2018)

Number 2

Chromatic classical symmetric functions

Pages: 401 – 409

DOI: https://dx.doi.org/10.4310/JOC.2018.v9.n2.a8

Authors

Soojin Cho (Department of Mathematics, Ajou University, Suwon, South Korea)

Stephanie van Willigenburg (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)

Abstract

In this note we classify when a skew Schur function is a positive linear combination of power sum symmetric functions. We then use this to determine precisely when any scalar multiple of a skew Schur function is the chromatic symmetric function of some graph. From here we are able to prove that of the classical bases for symmetric functions only certain scalar multiples of the elementary symmetric functions can be realised as the chromatic symmetric function of some graph, namely a particular union of complete graphs.

Keywords

chromatic symmetric function, complete graph, elementary symmetric function, skew Schur function

2010 Mathematics Subject Classification

Primary 05E05. Secondary 05C15, 05C25.

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The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01057476).

The second author was supported in part by the National Sciences and Engineering Research Council of Canada.

Received 4 January 2017

Published 22 January 2018