Rota, Gian-Carlo


Rota's basis conjecture ★★★

Author(s): Rota

Conjecture   Let $ V $ be a vector space of dimension $ n $ and let $ B_1,\ldots,B_n \subseteq V $ be bases. Then there exist $ n $ disjoint transversals of $ B_1,\ldots,B_n $ each of which is a base.

Keywords: base; latin square; linear algebra; matroid; transversal

Rota's unimodal conjecture ★★★

Author(s): Rota

Let $ M $ be a matroid of rank $ r $, and for $ 0 \le i \le r $ let $ w_i $ be the number of closed sets of rank $ i $.

Conjecture   $ w_0,w_1,\ldots,w_r $ is unimodal.
Conjecture   $ w_0,w_1,\ldots,w_r $ is log-concave.

Keywords: flat; log-concave; matroid

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