Partitions and Ferrers diagrams

A partition of a nonnegative integer n is a nonincreasing finite list of positive integers summing to n. The empty list is the unique partition of zero. The list $(4,2,2)$ is an integer partition of eight; changing its order does not produce another partition. A composition keeps order, so $(2,4,2)$ is different as a composition. A set partition is different again: it divides distinct labels into blocks, and block membership contains information absent from a list of integer sizes.

For each permitted positive size j, specify a nonnegative multiplicity. This produces a unique integer partition after sorting the parts. Allowing arbitrary multiplicity gives a geometric factor; allowing a part at most once gives a two-term factor. Although the unrestricted product contains infinitely many factors, its coefficient at degree n depends only on sizes up to n. That finite dependence justifies the product as a formal series without invoking convergence at a numerical point.

Draw a row of dots or cells for each part, left aligned with the longest row first. The conjugate partition has one part for every column, equal to the height of that column. Transposing twice restores the original diagram, proving bijectivity. The number of rows becomes the largest part and the largest part becomes the number of rows. Total size is preserved because the same cells are counted in a different direction.

Consequently partitions of n with at most k parts are equinumerous with partitions of n whose largest part is at most k. Equality conditions also transpose: exactly k rows correspond to largest part exactly k. These statements do not say that the two constrained families contain the same lists. For instance a long single row becomes a tall column. The diagram supplies the inverse map, which is stronger evidence than matching a short table of counts.

Partitions into distinct parts have the same count as partitions into odd parts. One formal derivation replaces each distinct-part factor by a quotient of two geometric numerators. Every even-indexed factor cancels, leaving reciprocal factors only for odd sizes. To interpret an infinite cancellation safely, work to a fixed degree and ignore all higher-degree terms. Only finitely many factors matter there, so the cancellation is an ordinary finite algebraic calculation at every requested coefficient.

There is also a reversible arithmetic explanation: in a partition into odd parts, express the multiplicity of each odd size in binary. Each nonzero binary digit becomes one part of that odd size multiplied by a power of two. The resulting parts are distinct because every positive integer has a unique odd factor times a power of two. Splitting each distinct part back into its odd factor recovers the original multiplicities. This bijection changes the number of parts, so it proves a total-size identity rather than a fixed-length identity.