  The study of asymptotic minimum degree thresholds that force matchings and
tilings in hypergraphs is a lively area of research in combinatorics. A key
breakthrough in this area was a result of H\`{a}n, Person and Schacht who
proved that the asymptotic minimum vertex degree threshold for a perfect
matching in an $n$-vertex $3$-graph is
$\left(\frac{5}{9}+o(1)\right)\binom{n}{2}$. In this paper we improve on this
result, giving a family of degree sequence results, all of which imply the
result of H\`{a}n, Person and Schacht, and additionally allow one third of the
vertices to have degree $\frac{1}{9}\binom{n}{2}$ below this threshold.
Furthermore, we show that this result is, in some sense, tight.
