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awarded  Popular Question 
1d

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Identities for power series like $\sum_n z^{n^3}$
In my book Enumerative Combinatorics*, vol. 2, solution to Exercise 6.63(c), I suggest that the series $\sum_{n=0}^\infty z^{n^3}$ does not satisfy an algebraic differential equation. This would rule out a wide class of possible identities. 
Mar 27 
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About structure of the set of perfect matchings of $K_{n,n}$
The matching polytope of $K_{n,n}$ leads to a "global structure" on the set of all matchings, e.g., those sets of matchings that form a face. 
Mar 26 
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Positivity of Ehrhart polynomial coefficients
Another class of polytopes whose Ehrhart polynomials have nonnegative coefficients are the latticeface polytopes of Fu Liu. See arxiv.org/abs/0810.4655. In particular, if $\mathcal{P}$ is any rational polytope, then there exists an integral polytope $\mathcal{Q}$ combinatorially equivalent to $\mathcal{P}$ such that the Ehrhart polynomial of $\mathcal{Q}$ has nonnegative coefficients. 
Mar 23 
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Communal problem books
The Caltech math library had such a book. I don't know whether it still exists. 
Mar 22 
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Enumeration of $01$ matrices with determinant $1$
I am guessing that most of the contribution comes from upper unitriangular matrices (though maybe this is nonsense). Why should the determinant distribution be approximately uniform in the range $[n^{n/2},n^{n/2}]$? On the other hand, the mean of $\mathrm{det}(A)^2$ is $4^{n}(n+1)!$. 
Mar 22 
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Enumeration of $01$ matrices with determinant $1$
Is it true that $f(n)=2^{\frac{n^2}{2}+o(n^2)}$? 
Mar 21 
awarded  Necromancer 
Mar 21 
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Positivity of Ehrhart polynomial coefficients
The zeros of the Ehrhart polynomial $H_n(k)$ of the Birkhoff polytope of $n\times n$ doubly stochastic matrices look very interesting. For $9\times 9$ matrices see math.mit.edu/~rstan/zeros/magic9.pdf. Assuming that this behavior generalizes to all $n$, we would have the following. Let $c(n,i)$ be the coefficient of $k^{(n1)^2i}$ in $H_n(k)$. Then for fixed $k$, $c(n,i)/c(n,0)\sim n^{3i}/2^ii!$. See EC1, 2nd ed., Exercise~4.54. 
Mar 20 
answered  Positivity of Ehrhart polynomial coefficients 
Mar 20 
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Positivity of Ehrhart polynomial coefficients
@PerAlexandersson: yow, you are right! I have made this mistake before. If $\Omega_P(k)$ is the order polynomial of a poset $P$, then $\Omega_P(k+1)$ (not $\Omega_P(k)$) is the Ehrhart polynomial of the order polytope. However, for any $n\geq 1$, the order polytope of the poset $P_n$ with one minimal element covered by $n$ other elements is $\sum_{i=1}^{k+1}i^n$. For $n=20$ the coefficient of $k$ is $168011/330$, so Ehrhart polynomials of 0/1 polytopes need not have nonnegative coefficients. 
Mar 20 
revised 
Are there dense sets of positive but not full measure?
sentence added at end 
Mar 20 
answered  Are there dense sets of positive but not full measure? 
Mar 20 
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Are there dense sets of positive but not full measure?
The second example of KjosHanssen does contain an interval. For an example not containing an interval, simply take $A=([0,1/2]\mathbb{Q})\cup([1/2,0]\cap \mathbb{Q})$. However, I am guessing that the condition you actually want is that for all $0\leq a<b\leq 1$, we have $0<\mu([a,b]\cap A)<ba$. 
Mar 19 
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Famous examples of PhD advisors younger than their student
Anders Björner simultaneously had both father and son as students: Henrik and Kimmo Eriksson. 
Mar 18 
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Positivity of Ehrhart polynomial coefficients
The Ehrhart polynomial of an order polytope can have negative coefficients. See EC1, 2nd ed., Exericse 3.164. Ehrhart polynomials of integer zonotopes do have nonnegative coefficients (e.g., Theorem 2.2 of math.mit.edu/~rstan/pubs/pubfiles/83.pdf). I am not sure if it's known whether or not the generalized permutohedra of Postnikov (math.mit.edu/~apost/papers/permutohedron_full.pdf) have this property. 
Mar 17 
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Does a polytope have a selfindexing shelling?
This question is a strong version (because continuous deformations are not allowed) of the dual of mathoverflow.net/questions/199357. 
Mar 17 
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Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?
@GeoffRobinson: yow! I meant congruent to 2 mod 4, not 0 mod 4. 
Mar 17 
answered  Generalized expression for balls and bins problem 
Mar 17 
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Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?
If $S$ consists of primes congruent to 0 or 1 mod 4, then the density is 0. This is because every element of span$(S)$ is a sum of two squares, and the set of positive integers that are a sum of two squares has density 0. In fact, we can throw into $S$ the squares of all primes congruent to 3 mod 4, and the density is still 0. 