Characterizing the supernorm partition statistic

Abstract

AbstractThe paper gives characterizations of the supernorm statistic $$\widehat{N} :\mathcal {P}\rightarrow \mathbb {N}^{+}$$ N ^ : P → N + for partitions, where $$\mathcal {P}$$ P is the set of integer partitions and $$\mathbb {N}^{+}$$ N + is the positive integers. The supernorm statistic map is a bijection onto $$\mathbb {N}^{+}$$ N + that defines a total order on partitions, the supernorm ordering, obtained by pulling back the additive total order on $$\mathbb {N}^{+}$$ N + . The supernorm ordering refines two partial orders on partitions: the multiset inclusion order, whose image under $$\widehat{N}$$ N ^ is the divisibility lattice on $$\mathbb {N}^{+}$$ N + , and the Young’s lattice order. The paper shows that it is characterized by these two properties, additionally with the requirement of an order-isomorphism from the multiset inclusion order to the divisibility order. It is also characterized by these two properties, additionally with the requirement of mapping the partitions with exactly one part bijectively to the prime numbers. It presents a construction showing that the latter additional conditions are necessary for the characterization to hold.