Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra

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The asymptotic volume of the Birkhoff polytope (2007, methodology): “In particular, we give an asymptotic formula for the volume of Bn.”
Flow polytopes of signed graphs and the Kostant partition function (2012, background): “If P has (rational) integral vertices then LP(t) is a (quasi) polynomial (for background on the theory of Ehrhart polynomials see [6]).”
A generating function for all semi-magic squares and the volume of the Birkhoff polytope (2007, background): “Furthermore, the leading coefficient of e(P, t) is the normalized volume of P in units equal to the volume of the fundamental domain of the affine lattice spanned by P (see Chapter 4 of [22] or the book [7]).”
An Update on the Hirsch Conjecture (2009, background): “Last but not least, counting integer points in polytopes with integer vertex coordinates has applications ranging from number theory and representation theory to cryptography, integer programming, and statistics [5, 16].”
Positivity theorems for solid-angle polynomials (2009, background): “The interested reader can find proofs and much more in [BR].”
The 3x+1 Problem: An Overview (2021, methodology): “In another direction, functions that count the number of lattice points inside dilated rational polyhedra have been shown to be quasi-polynomial functions (on the positive integers), starting with work of Ehrhart [23], see Beck and Robins [6] and Barvinok [5, Chap.”