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Abstract: I present a bijection on integer partitions that leads to recursiveexpressions, closed formulae and generating functions for the cardinality ofcertain sets of partitions of a positive integer $n$. The bijection leads alsoto a product on partitions that is associative with a natural grading thusdefining a free associative algebra on the set of integer partitions. As anoutcome of the computations, certain sets of integers appear that I calldifference sets and the product of the integers in a difference set is aninvariant for a family of sets of partitions. The main combinatorial objectsused in these constructions are the central hooks of the Ferrers diagrams ofpartitions.

Author: Alain Goupil

Source: https://arxiv.org/

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