The Whitney numbers of a matroid encode the numbers and incidence structure of its flats. For a loopless
matroid 
of matroid rank 
on an 
-element ground set, the Whitney
number of the second kind 
is the number of flats of
matroid rank 
. The unsigned Whitney numbers of the first kind 
are defined by
 |
where 
is the ground set, 
is matroid rank, and 
is the characteristic polynomial
of the matroid (Dong and Liu 2026).
Dong and Liu (2026) proved Brylawski's conjecture that the sequence 
is a logarithmically
concave sequence. The normalization uses the size 
of the ground set, not the matroid rank 
. They also proved the points-lines-planes inequality 
for matroids
of matroid rank at least three, together with a sharper
version and equality characterizations. This does not assert that the entire sequence
is a logarithmically
concave sequence, the different, refuted Mason's
conjecture on flats.
Dong and Liu (2026) report AI assistance with mathematical exploration and technical checking.
