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awarded  Popular Question
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comment 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
comment 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
comment Positivity of Ehrhart polynomial coefficients
Another class of polytopes whose Ehrhart polynomials have nonnegative coefficients are the lattice-face 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
comment Communal problem books
The Caltech math library had such a book. I don't know whether it still exists.
Mar
22
comment Enumeration of $0-1$ 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
comment Enumeration of $0-1$ matrices with determinant $1$
Is it true that $f(n)=2^{\frac{n^2}{2}+o(n^2)}$?
Mar
21
awarded  Necromancer
Mar
21
comment 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 $k^{(n-1)^2-i}$ 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
comment 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
comment Are there dense sets of positive but not full measure?
The second example of Kjos-Hanssen 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 $0<\mu([a,b]\cap A)<b-a$.
Mar
19
comment 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
comment 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
comment Does a polytope have a self-indexing shelling?
This question is a strong version (because continuous deformations are not allowed) of the dual of mathoverflow.net/questions/199357.
Mar
17
comment 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
comment 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.