Not much, but the new thread is https://polymathprojects.org/2017/03/06/rotas-basis-conjecture-polymath-12-2/.

]]>I think I managed to derive two formulas (I’m not sure if they will be of use):

For , we have where is the set of -matrices, is the number of ‘s in , and is the matrix formed by deleting the -th column of .

For , we have where is the set of -matrices, is the number of ‘s in , and is the matrix of minors satisfying for all .

Since the proofs are cumbersome and my WordPress LaTeX skills are not great, I wrote up what I’ve done at Overleaf: https://www.overleaf.com/8425738kswcbnfsfzqw

]]>Daniel Kotlar pointed out that this observation is true for even n>=2 is Theorem 3.3 in Aharoni and Loebl, “The odd case of Rota’s bases conjecture” (2015). http://dx.doi.org/10.1016/j.aim.2015.06.015

]]>This is fascinating! In these small terms also divides for every . But I suppose we know from Glynn’s result that 13 that divides does not divide .

]]>Let denote the number of even Latin squares with the first row and column in order, and denote the number of odd Latin squares with the first row and column in order. The values of are when . [Stones and Wanless (2012)]

I just stumbled upon the surprising property: for , this sequence has the property that What’s up with that?! I find it hard to believe this is just a small-case numerical coincidence, since when we have which are some big numbers.

In Stones and Wanless (2012), we conjectured for all n. If the above observation is true for all even n, then the Alon-Tarsi Conjecture also implies our conjecture. I.e., the two conjectures are equivalent.

]]>Yes, it looks like you are right. The topological method does not give what I thought it would.

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