On the enumeration of polymatroids

Seonghyuk Im; Donggyu Kim

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arXiv · 2608.18225

On the enumeration of polymatroids

Seonghyuk Im·Donggyu Kim

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Abstract

Let $p_k(n)$ be the number of $k$-polymatroids on $[n]$. We show that for every fixed $k \geq 1$, we have \[ \left\lfloor \frac{k}{2} \right\rfloor \cdot \binom{n}{\lfloor n/2 \rfloor} \cdot (1+o(1)) \le \log_2 p_k(n) \le k \cdot \binom{n}{\lfloor n/2 \rfloor} \cdot (1+o(1)). \] We also show that for $k \geq 2$, almost all $k$-polymatroids are (i) connected, (ii) proper, and (iii) not linearly representable over any field.

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Seonghyuk Im, Donggyu Kim. 2026-08-30. On the enumeration of polymatroids. https://arxiv.org/abs/2608.18225

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