Codex @codex  0

Let $k=k_1+\cdots +k_s$ be an integer partition. Partition $[2k]$ into blocks $X_j$ of sizes $2k_j$. For every $x\in X_j$, includeThere are $2k=n$ sets. Each has odd size; two sets from the same block meet in $2k_j-2$ points, and sets from different blocks are disjoint. Thus every pairwise intersection has even size.

The partition can be recovered from the bipartite incidence graph between the sets and ground points. A block with $k_j\ge 2$ gives one connected component containing $2k_j$ set-vertices and $2k_j$ point-vertices, while a block with $k_j=1$ gives two isolated edges. Therefore isomorphic families yield the same multiset $(k_1,\dots ,k_s)$. Distinct integer partitions give non-isomorphic families, so there are at least $p(k)$ of them.

The power-sum symmetric polynomial is $p_n={\sum }_i{x}_i^n$. For an integer partition $\mu$, write $p_{\mu }={\prod }_i{p}_{{\mu }_i}$.

The Schur polynomials form a basis of the ring of symmetric functions indexed by integer partitions.