Geometrical circuits and matroids? A002631 and A002632

Gordon Royle gordon at
Thu Sep 30 11:06:44 CEST 2004

Dear Seqfanners..

I have been doing some computing with binary matroids, and in 
particular I constructed the regular matroids of rank up to 6, and then 
separated out from them the graphic ones (cycle matroid of a graph) and 
the cographic ones (bond matroid of a graph).

I got the following numbers for ranks 1..6

Regular: 1, 2, 5, 17, 81, 634
Graphic: 1, 2, 5, 16, 73, 533
Cographic: 1, 2, 5, 16, 67, 435

I then put them into OEIS to see what came out...

For regular matroids, I got nothing.

For graphic matroids, I got a close match to sequence A002632 which is 
entitled "Geometrical circuits of rank n" and has a reference to a 1932 
paper by R.M. Foster - this sequence started 1, 2, 5, 16, 73, but then 
had next (and final) term equal to 538... maybe a typo or a mistake or 
maybe the squences are just not related..

For cographic matroids, I got an EXACT match to sequence A002631 which 
is entitled "Number of circuits of nullity n"...

So, can anyone enlighten me as to what "geometrical circuits" are, and 
why (or whether) they actually are related to graphic and/or cographic 

(The numbers of graphic matroids of rank n should be the numbers of 
connected graphs on n+1 vertices EXCEPT that any two graphs with the 
same blocks will only contribute 1 to the total count because they lead 
to isomorphic matroids...)



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