Background
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What I think of *Ehrhart theory* (<http://en.wikipedia.org/wiki/Ehrhart_polynomial>) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th dilation of $P$, the result is polynomial in $t$.


If, however, $P$ is only a *rational* polytope, then in general we get a quasi-polynomial; that is, there is some period $N$, so that if for any $k$, I only look at $t$ congruent to $k$ mod $N$, then again I have a polynomial behavior; however, different residue classes will in general result in different polynomials. 


One might expect that any two distinct residue classes have distinct polynomials, but this need not be the case: two different residue classes can have the same polynomial. These are the "accidental" equalities of the question title.


A simple example
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For a simple example, take the triangle bounded by $x=0,y=3x$ and $y=1$; scaling by $t$ just changes $y=1$ to $y=t$. The count is quasi-polynomial of period 3, but only has two distinct polynomials. The number of lattice points is 


$$ 1+\frac{t^2+5t}{6}$$
if $t$ is congruent to 0 or 1 mod 3, and
$$ 1+ \frac{t^2+5t-2}{6}$$
if $t$ is congruent to 2 mod 3.


Questions, general and specific
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I'm interested in what we know about when or why these accidental equalities occur. That is a rather broad and open-ended question; so here's something a bit more specific.


If instead of just counting the lattice points in the $t$th dilate, we sum a polynomial function over the lattice points, we again get (quasi-)polynomial functions; call this *Euler-Maclaurin* theory. I had the naive hope that "Accidental" equalities in Ehrhart quasi-polynomials might extend to equalities in Euler-Maclaurin theory, but this appears not be always the case: if we try to sum the function $x$ over the lattice points in the $t$th dilate of the polytope in our example, we get
$$\frac{t^3}{54}+\frac{t^2}{9}+\frac{t}{6}$$
for $t$ congruent to 0 mod 3, but 
$$\frac{t^3}{54}+\frac{t^2}{9}+\frac{t}{18}-\frac{5}{27}$$
for $t$ congruent to 1 mod 3. 


If I have some rational polytope that has accidental equalities in its Ehrhart polynomials, and a specific polynomial I want to sum over it; is there some conditions in which the same accidental equalities will hold for the Euler-Maclaurin problem?



How "accidental" are equalities between parts of Ehrhart quasi-polynomials? When do they persist to Euler-Maclaurin?
This is an interesting and wide open question. The easiest case when considering Ehrhart quasipolynomials, namely, when *all* constituent polynomials are equal, goes by the name of *period collapse*. The most famous instance comes from representation theory (see [this paper by J. De Loera & T. McAllister](http://arxiv.org/pdf/math.CO/0309329.pdf), which appeared in Discrete Comp. Geom. 32 (2004)). You might also find [this paper by C. Haase & T. McAllister](http://arxiv.org/pdf/0709.4070.pdf) (which appeared in Cont. Math. 451 (2008)) illuminating.


