Beta number of a partition (${\beta }_i={\lambda }_i+l-i$)

For a partition padded to $l$ rows, its beta numbers are the distinct nonnegative integers ${\lambda }_i+l-i$. They are its occupied abacus of a partition positions. With exactly the positive row count, they equal the first-column hook lengths; extra zero rows change the representation of the beta set while preserving the partition. Individual beta numbers describe bead positions, whereas the beta set is the collection of all these numbers.