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Let $M_{m,n}(\Bbb{C})$ be the space of $m\times n$ matrices with entries in $\Bbb{C}$, and let $U_{k,m.n}(\Bbb{C})\subset M_{m,n}(\Bbb{C})$ be the variety of matrices of rank $\leq k\leq\min(m,n)$. Then the projectivization $\Bbb{P}U_{k,m,n}(\Bbb{C})$ is an irreducible variety of codimension $(m-k)(n-k)$ in $\Bbb{P}M_{m,n}(\Bbb{C})$. Note that $\Bbb{P}U_{k-1,m,n}$ is the variety of singular points of $\Bbb{P}U_{k,m,n}$.

In this paper, Harris and Tu computed the degree of $\Bbb{P}U_{k,m,n}(\Bbb{C})$ as $$\gamma_{k,m,n}:=\deg\Bbb{P}U_{k,m,n}(\Bbb{C})=\prod_{j=0}^{n-k-1}\frac{\binom{m+j}{m-k}}{\binom{m-k+j}{m-k}}.$$ If we write $a=n-k, b= m-k, c=k$ and introduce $$\psi(a,b,c):=\prod_{j=0}^{a-1}\frac{\binom{b+c+j}b}{\binom{b+j}b},$$ then $\gamma_{k,m,n}=\psi(a,b,c)$. Incidentally, $\psi(a,b,c)$ enumerates plane partitions contained in an $a\times b\times c$ rectangular box.

In a follow up paper, Friedland and Krattenthaler investigated (among other related numbers) the parity of $\gamma_{k,m,n}$. However, their characterization involves a long list of case-study (see, pages 22-26), also there is a commentary by the authors on page 3:

"We also consider the problem of characterizing the values of $k,m,n$ for which $\gamma_{k,m,n}$ is odd. This problem seems to have a rather intricate solution. We give some partial results on this problem in Section 6."

Given an integer $j$, let $s(j)=$ the sum of the binary digits of $j$ (e.g. $s(7)=s(111_2)=3$ and $s(8)=(1000_2)=1$). Denote $\Phi(u,v)=\sum_{j=u}^vs(j)$. Then, it is not hard to show that $\psi(a,b,c)$ is odd iff $$\Phi(a+b,a+b+c-1)+\Phi(0,c-1)=\Phi(a,a+c-1)+\Phi(b,b+c-1).$$

I've found a compact formulation that seems to work (experimentally) in all cases and thereby possibly amenable to a shorter proof, too.

Question 1. Can you prove that if $$\text{either $\,\,\,\,n\equiv k \mod 2^{\lceil{\log_2m}\rceil}\,\,\,\,\,\,$ or $\,\,\,\,\,\,n+m\equiv k \mod 2^{\lceil{\log_2m}\rceil}$}$$ then $\gamma_{k,m,n}$ is an odd integer?

UPDATE. After checking what Gjergji Zaimi pointed out, I realized that I made errors in my calculations. So, I have modified the question (instead of "iff" it is now "if").

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    $\begingroup$ Hm, is this condition symmetric in $a,b,c$? $\endgroup$ May 17, 2017 at 20:02
  • $\begingroup$ It should be, although I did not check. $\endgroup$ May 17, 2017 at 20:11
  • $\begingroup$ "Either $a$ or $a+b+c$ is divisible by the minimal power of 2 which is not less than $c$." $\endgroup$ May 17, 2017 at 20:46
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    $\begingroup$ If I understand correctly I get $\gamma_{8,10,10}$ to be odd, but your condition says it should be even. $\endgroup$ May 17, 2017 at 21:49
  • $\begingroup$ @GjergjiZaimi: Thank you, you're right, I made mistakes there. Please see the update question. $\endgroup$ May 18, 2017 at 11:41

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