Higher Order Bell Symmetric Functions

Abstract: We study symmetric function analogues of the higher order Bell numbers. Their construction involves iterated plethystic exponential towers mimicking the single variable exponential generating functions for the higher order Bell numbers. We derive explicit recurrence relations for the expansion coefficients of the Bell functions into the monomial and power sum bases of the ring of symmetric functions. Using the machinery of combinatorial species, the Bell functions are proven to be the Frobenius characteristics of the permutation representations of symmetric groups on hyper-partitions of certain orders and sizes. In the order 1 case, we are able to give more details about the expansion coefficients of the Bell functions in terms of vector partitions and divisor sums as well as give a recurrence relation analogous to the well known recursion for the Bell numbers. Lastly, we use Littlewood's reciprocity theorem and the Hardy-Littlewood Tauberian theorem to prove that the Schur expansion coefficients of the order 1 Bell functions are certain asymptotic averages of restriction coefficients.