  The emergence of Boij-S\"oderberg theory has given rise to new connections
between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro
recently showed that every Betti diagram of an ideal with a k-linear minimal
resolution arises from that of the Stanley-Reisner ideal of a simplicial
complex. In this paper, we extend their result for the special case of 2-linear
resolutions using purely combinatorial methods. Specifically, we show bijective
correspondences between Betti diagrams of ideals with 2-linear resolutions,
threshold graphs, and anti-lecture hall compositions. Moreover, we prove that
any Betti diagram of a module with a 2-linear resolution is realized by a
direct sum of Stanley-Reisner rings associated to threshold graphs. Our key
observation is that these objects are the lattice points in a normal reflexive
lattice polytope.
