##### This material has been published in
Linear
Algebra Appl. **426** (2007), 159-189,
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# 2-adic valuations of certain ratios of products of
factorials and applications

### (32 pages)

**Abstract.**
We prove the conjecture of Falikman-Friedland-Loewy
on the parity of the degrees of projective varieties of
*n*x*n* complex symmetric matrices of rank at most *k*.
We also characterize the parity of the degrees of projective
varieties of *n*x
*n* complex skew symmetric matrices of rank at most 2*p*.
We give recursive relations which determine the parity of the
degrees of projective varieties of *m*x*n* complex matrices
of rank at most *k*.
In the case the degrees of these varieties are odd, we characterize
the minimal dimensions of subspaces of *n*x
*n* skew symmetric real matrices and of *m*x*n*
real matrices containing a nonzero matrix of rank at most *k*.
The parity questions studied here are also of combinatorial interest
since they concern the parity of the number of plane partitions
contained in a given box, on the one hand, and the parity of
the number of symplectic tableaux of rectangular shape, on the
other hand.

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