Geometrical circuits and matroids? A002631 and A002632
Gordon Royle
gordon at csse.uwa.edu.au
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
matroids?
(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...)
Thanks..
Gordon
More information about the SeqFan
mailing list