Alternate transition matrices for Brentiʹs image-symmetric functions and a class of image-symmetric functions on the hyperoctahedral group Original Research Article

Recently, Brenti introduced a class of image-symmetric functions based on a simple plethysm with the power-sum symmetric functions. Brenti developed combinatorial interpretations for the transition matrices between these new symmetric functions and the standard symmetric function bases. We provide simplified versions of many of these that are sums over significantly smaller classes of combinatorial objects. We also show that two of our results generalize formulas of MacMahon relating vector compositions to the expansion of a product of binomial coefficients as a sum of binomial coefficients. We then extend Brentiʹs definitions to symmetric functions on the hyperoctahedral group, image, and give combinatorial interpretations of the analogous transition matrices. We also discuss new generating functions on permutation statistics that arise from Brentiʹs symmetric functions and our extensions, two of which show a curious connection to Stirling numbers of the second kind.