The lattice of set partitions and transition matrices of symmetric functions

Block tabloids are a combinatorial object introduced by O. Egecioglu and J. Remmel through which the transition matrices between the bases m,e,p and h of the commuting symmetric functions may be defined. There is an alternative way to describe the transition matrices using symmetric functions in non-commuting variables and the lattice of set partitions. Our goal is to study functions on the lattice of set partitions that arise as entries in the transition matrices. Our research explores the relationship between brick tabloids and functions on the lattice of set partitions. For example, we study Nλ(μ), the number of set partitions of type μ that are larger than or equal to a set partition of type λ, and nλ(μ), the number of set partitions of type μ that are less than or equal to a set partition of type λ. 

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