Affiliation:
1. Department of Mathematics, University of British Columbia , Vancouver BC V6T 1Z2, Canada
2. School of Mathematical Sciences, Dalian University of Technology , Dalian Liaoning 116024, P.R. China
Abstract
Abstract
We define vertex-colourings for edge-partitioned digraphs, which unify the theory of $P$-partitions and proper vertex-colourings of graphs. We use our vertex-colourings to define generalized chromatic functions, which merge the chromatic symmetric and quasisymmetric functions of graphs and generating functions of $P$-partitions. Moreover, numerous classical bases of symmetric and quasisymmetric functions, both in commuting and noncommuting variables, can be realized as special cases of our generalized chromatic functions. We also establish product and coproduct formulas for our functions. Additionally, we construct the new Hopf algebra of $r$-quasisymmetric functions in noncommuting variables, and apply our functions to confirm its Hopf structure, and establish natural bases for it.
Publisher
Oxford University Press (OUP)