I am solving some problem related to symmetric matrices over a finite field $\mathbb{F}_q$ and I am stuck at the following problem: For every $a\in\mathbb{F}_q $, let $S_a(t,m)$ be the set of all $m\times m$ symmetric matrices over $\mathbb{F}_q$ of rank $t$ and trace $a$. Then I want to prove that $|S_0(t, m)|\geq |S_1(t, m)|$( here $1$ is not special. it could be any nonzero element of the field). I have spent lots of time trying to get either an injection from $S_1(t, m)$ to $S_0(t, m)$ or a surjection in other direction but could not succeed. Some calculations in SAGE shows it is correct but have no idea how should I proceed. Any help or reference is appreciated. Is it possible to prove this using some probability theory?
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$\begingroup$ Perhaps the paper projecteuclid.org/euclid.dmj/1077380573 can be extended to symmetric matrices. $\endgroup$– Richard StanleyCommented Apr 17, 2018 at 21:59
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$\begingroup$ Thanks @ Richard Stanley for the reference. The reference is not available at my university but I am trying to find it. Although, here I don't want to count them but to prove that inequality. Counting could be more complex as it is for general matrices(it is mentioned in the review of the paper.) $\endgroup$– SinghCommented Apr 18, 2018 at 10:00
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