Abstract

There is a natural shift action defined on $r$-partitions with $r \geq 2$. This action is compatible with another shift action on associated multisets of residus (or ‘blocks’). We proved, using a constrained quadratic optimisation problem on integers, that the side of the orbit of a block coincides with the size of the orbit of at least one $r$-partition in this block. In a recent work, we studied the case $r = 1$. In this case, the shift operation on blocks is still defined but not the one on partitions. Via a natural generalisation of the weight function for partitions, we prove that the set of all blocks can be obtained as a superlevel set for this generalised weight function. We then obtain a necessary and sufficient condition so that the shift of a block corresponds to a partition, in which case we construct such a partition. We will see that the theorem between orbit sizes for $r \geq 2$ remains true in some cases. Finally, we will see how we used the notion of core block, as introduced by Fayers, to prove that the set of all blocks for $r$-partitions (with $r \geq 2$) contains a superlevel set for the generalised weight function.