Actually, I goofed last night: what I had in mind was a comparison of 101 000 101 ("before") and 101 010 101 ("after"). The intuition behind my example will be clearer if I explain these arrays as (grid-)multipaths. Label the nine vertices as ABCD EFGH JKLM NOPQ (I left out "I" because it's so narrow, since so many of us use non-fixed-width mail-readers). Start with a 4-component multipath consisting of the closed paths AEFBA, JNOKJ, LPQML, and CGHDC. Now add the four edges FK, KL, LG, and GF. We get the 1-component path AEFKJNOKLPQMLGHDCGFBA, and only the winding number around the center square has changed. (I hope I got it right this time!) Anyway, the multipath changes from having four components to having just one component. (Unless Tom and Gareth are using a different definition of components.) Anyway, in addition to wanting to understand better why there must be at least k-1 grid-squares at which single-component pattern doesn't achieve maximal winding number, I'd like to know the constraints on the locations of those grid-squares. In Allan's example, the grid-squares were all on the same diagonal. Is that forced by Tom and Gareth's analysis (which I'm still trying to understand)? Jim On Thu, Mar 24, 2016 at 7:03 AM, James Propp <jamespropp@gmail.com> wrote:
I expect that Gareth and Tom will find my remarks about
010 111 010
obtuse and irrelevant, since local configurations like that won't occur as one successively mutates the array (starting from the maximal array and lowering k-1 entries one at a time). But how do we KNOW they won't? Clearly Tom and Gareth have some additional intuitions about what's going on here that aren't captured by the analyses they've offered.
Jim
On Wednesday, March 23, 2016, James Propp <jamespropp@gmail.com> wrote:
Thanks for taking the time to clarify, Gareth.
4. Whatever that reduction is, we can imagine doing it
one square (and one unit of winding for that square) at a time.
This is not obvious to me, but I can prove it. (You have to decrement the larger numbers first.)
When we reduce one PSWN by 1, the number of path components cannot reduce by more than one -- for we cannot have components of more than two disconnected paths adjacent to a single square.
This lemma (if I'm understanding it correctly) cannot hold in full generality. Consider the PSWN array
010 101 010
(four separate loops) and the PSWN array
010 111 010
(a single loop); when we turn the former into the latter, a single PSWN changes by 1, but the number of components changes by 3.
I'll mention Allan Wechsler here, for no reason other than the fact that I seem to keep misspelling his first name and could use some practice spelling it right. Allan, Allan, Allan. There! Let's see if that helps. :-)
Jim