Conventions for integer partitions, Ferrers diagrams, and the twelvefold-way table

Throughout this page, $n$ counts balls or domain elements and $k$ counts boxes or codomain values.

When partitions are displayed on this page, they are written in nonincreasing order

${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_r>0,$

even though Partitions of a positive integer stores the same data as a nondecreasing list.

A composition of $n$ into $k$ parts means an ordered $k$-tuple of positive integers summing to $n$, while a weak composition allows zero parts. Thus Compositions of $n$ into $k$ positive parts are counted by $\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)$ counts positive occupancies and For $m\ge 1$ the number of weak compositions of $n$ into $m$ parts is $\left(\genfrac{}{}{0px}{}{n+m-1}{m-1}\right)$, and the number of compositions is $\left(\genfrac{}{}{0px}{}{n-1}{m-1}\right)$ for $n\ge 1$ counts arbitrary occupancies.

For placements of indistinguishable balls into boxes:

• if the boxes are labelled, the data are the occupancy vector $(a_1,\dots ,a_k)$;
• if the boxes are unlabelled, the data are the same occupancies reordered into nonincreasing order, with zero occupancies omitted.

Hence an unlabelled-to-unlabelled placement is encoded by a partition whose parts are the nonzero occupancies. Ferrers diagrams are read in English convention, with the longest row on top and rows left-justified.