A Complete Multipartite Basis for the Chromatic Symmetric Function
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Date
2021-11-15
Authors
Crew, Logan
Spirkl, Sophie
Journal Title
Journal ISSN
Volume Title
Publisher
Society for Industrial and Applied Mathematics
Abstract
In the vector space of symmetric functions, the elements of the basis of elementary symmetric functions are (up to a factor) the chromatic symmetric functions of disjoint unions of cliques. We consider their graph complements, the functions {rλ:λ an integer partition} defined as chromatic symmetric functions of complete multipartite graphs. This basis was first introduced by Penaguiao [J. Combin. Theory Ser. A, 175 (2020), 105258]. We provide a combinatorial interpretation for the coefficients of the change-of-basis formula between the rλ and the monomial symmetric functions, and we show that the coefficients of the chromatic and Tutte symmetric functions of a graph G when expanded in the r-basis enumerate certain intersections of partitions of V(G).
Description
First Published in SIAM Journal on Discrete Mathematics in volume 35, issue 4, 2021, published by the Society for Industrial and Applied Mathematics (SIAM)” and the copyright notice as stated in the article itself (e.g., “Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.”')
Keywords
chromatic symmetric function, symmetric functions, algebraic combinatorics