Asymptotic enumeration of hypergraphs by degree sequence

TL;DR

This paper derives an asymptotic formula for counting $k$-uniform hypergraphs with a specified degree sequence, extending previous graph results to hypergraphs and relating degree sequences to binomial models.

Contribution

It provides a new asymptotic enumeration formula for hypergraphs with given degree sequences, covering a broad parameter range and generalizing prior graph enumeration results.

Findings

Derived an asymptotic formula for hypergraph counts with given degree sequences.

Extended enumeration results from graphs to hypergraphs.

Linked degree sequences to binomial random variable models.

Abstract

We prove an asymptotic formula for the number of $k$-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that is asymptotically equal to the number of $d$-regular $k$-uniform hypergraphs on $n$ vertices provided that $dn\le c\binom{n}{k}$ for a constant $c>0$, and $3 \leq k < n^C$ for any $C<1/9.$ Our results relate the degree sequence of a random $k$-uniform hypergraph to a simple model of nearly independent binomial random variables, thus extending the recent results for graphs due to the second and third author.