Thank you. A matroid is sparse paving if both the matroid itself and its dual are paving. This criterion can be used to count the number of sparse paving matroids, assuming the .is_paving() method is correct. Sparse paving matroids have several definitions, but as far as I know, their definition is unambiguous.
在2025年1月27日星期一 UTC+8 02:24:58<dim...@gmail.com> 写道： > On Sat, Jan 25, 2025 at 11:03 PM Xie <xiehon...@gmail.com> wrote: > > > > for M in matroids.AllMatroids(8, type='sparse_paving'): > > ....: print(M) > > sparse_paving_n08_r04_#0: Matroid of rank 4 on 8 elements with 56 bases > > sparse_paving_n08_r05_#0: Matroid of rank 5 on 8 elements with 48 bases > > sparse_paving_n08_r06_#0: Matroid of rank 6 on 8 elements with 24 bases > > sparse_paving_n08_r07_#0: Matroid of rank 7 on 8 elements with 8 bases > > sparse_paving_n08_r07_#1: Matroid of rank 7 on 8 elements with 7 bases > > sparse_paving_n08_r08_#0: Matroid of rank 8 on 8 elements with 1 bases > > > > This can't be correct because almost all matroids are sparse paving. > > this has been conjectured to be held asymptotically (i.e. as # n of > elements goes to infinity) > in https://doi.org/10.1016/j.ejc.2011.01.016 > and https://www.sciencedirect.com/science/article/pii/S0196885812000802 > > With this in mind, it's hard to understand what exactly could be wrong > there (as this is something for n=8, not for n->oo) > > > > > > In SageMath, the is_sparse_paving‘s Docstring defines: > > > > *"Return if 'self' is sparse-paving. > > > > A matroid is sparse-paving if the symmetric difference of every pair of > circuits is greater than 2."* > > > > I believe this is incorrect! > > indeed, this seems strange, and no references are provided. > I've left a comment to this effect here: > https://github.com/sagemath/sage/pull/36962#issuecomment-2614537747 > > > > > -- > > You received this message because you are subscribed to the Google > Groups "sage-support" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to sage-support...@googlegroups.com. > > To view this discussion visit > https://groups.google.com/d/msgid/sage-support/a0f42d8e-a377-465e-b224-04166d86bc80n%40googlegroups.com > . > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/sage-support/25e1e973-4386-420f-86d3-d19003b86149n%40googlegroups.com.
