4 note another formulation

Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, equivalently the probability that a random matrix from the uniform distribution on $M(n,k)$ has no zero entries.

One thing to note is that $$P(n,k)=\frac{|M(n,k-n)|}{|M(n,k)|}$$ (think about subtracting 1 from every entry). Also note that $P(n,k)$ is the fraction of integer points in the $k$-dilated Birkhoff polytope that lie in the interior.

It seems "obvious" that $P(n,k)$ is a non-decreasing function of $k$. For large enough $k$ it is strictly increasing by Ehrhart theory, but I'd like to see a proof for all $k$. So the problem is:

Prove that $P(n,k)$ is a non-decreasing function of $k$ for fixed $n$.

COMMENT: By a result of Stanley (see David Speyer's answer for refs) there are non-negative integers $\{h_i\}$ such that $$|M(n,k)| = \sum_{i=0}^d h_i \binom{k+i}{d}$$ where $d=(n-1)^2$. I'm wondering if this is enough. Does every polynomial of that form have the desired properties? If so, the result will generalise [Gjergji showed not.]

COMMENT2: The reciprocity theorem for Ehrhart series provides a lot since every lattice formula for the number of points in the interior in terms of the number in the whole (closed) polytopehas such an enumerator. Careful: some Making use of the binomial coefficients are 0 when above, we find that if the $k\lt d$H_n(k)$is the polynomial equal to$|M(n,k)|$for positive integers$k$, then the number of interior points (already identified as$H_n(k-n)$) equals$(-1)^{n+1}H_n(-k)$. So what we have to prove is that $$(-1)^{n+1}\frac{H_n(-k)}{H_n(k)}$$ is non-decreasing for integer$k\ge n$. Experimentally, it is not increasing for real$k$until$k$is larger. 3 oops Let$M(n,k)$be the set of$n\times n$matrices of nonnegative integers such that every row and every column sums to$k$. Let$P(n,k)$be the fraction of such matrices which have no zero entries, equivalently the probability that a random matrix from the uniform distribution on$M(n,k)$has no zero entries. One thing to note is that $$P(n,k)=\frac{|M(n,k-n)|}{|M(n,k)|}$$ (think about subtracting 1 from every entry). Also note that$P(n,k)$is the fraction of integer points in the$k$-dilated Birkhoff polytope that lie in the interior. It seems "obvious" that$P(n,k)$is a non-decreasing function of$k$. For large enough$k$it is strictly increasing by Ehrhart theory, but I'd like to see a proof for all$k$. So the problem is: Prove that$P(n,k)$is a non-decreasing function of$k$for fixed$n$. COMMENT: By a result of Stanley (see David Speyer's answer for refs) there are non-negative integers $\{h_i\}$ such that $$|M(n,k)| = \sum_{i=0}^d h_i \binom{k+i}{d}$$ where$d=(n-1)^2$. I'm wondering if this is enough. Does every polynomial of that form have the desired properties? If so, the result will generalise a lot since every lattice polytope has such an enumerator. Careful: some of the binomial coefficients are 0 when$k\lt d$. 2 note polytope version of problem Let$M(n,k)$be the set of$n\times n$matrices of nonnegative integers such that every row and every column sums to$k$. Let$P(n,k)$be the fraction of such matrices which have no zero entries, equivalently the probability that a random matrix from the uniform distribution on$M(n,k)$has no zero entries. One thing to note is that $$P(n,k)=\frac{|M(n,k-n)|}{|M(n,k)|}$$ (think about subtracting 1 from every entry). Also note that$P(n,k)$is the fraction of integer points in the$k$-dilated Birkhoff polytope that lie in the interior. It seems "obvious" that$P(n,k)$is a non-decreasing function of$k$. For large enough$k$it is strictly increasing by Ehrhart theory, but I'd like to see a proof for all$k$. So the problem is: Prove that$P(n,k)$is a non-decreasing function of$k$for fixed$n\$.

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