Fano plane

Fano support:

This support S comes form the incidence matrix of the Fano plane (coming from the point-line incidence graph of the Fano plane, known as the Heawood graph). 

Since the total number of matrices (all ranks) is q^21, this implies that there exists a rank r=0,1,2...,6 such that the number of 7-by-7 rank r matrices over Fq with support in S is not a polynomial in q. Using the  procedure matqBruhat we computed these numbers (see FanoAllRanks.txt). For r=0,1,2,3 we obtain a polynomial in q and for r=4,5,6 we obtain two polynomials depending on the parity of q.

For the invertible case, the Fano example of Stembridge was the only known example of a support S such that the number of invertible matrices over a finite field with support contained in S is not a polynomial in q. Using the procedure matqBruhat we found two other non polynomial examples: the complement of the Fano plane:

, and the Fano plane embedded in a 14-by-14 matrix.

[1] J.R. Stembridge, Counting points on varieties over finite fields related to a conjecture of Kontsevich. Ann. Comb., 2(4):365–385, 1998. preprint