# [seqfan] Re: Partition into Stroke

Allan Wechsler acwacw at gmail.com
Sat Jan 29 00:15:25 CET 2022

```Yasutoshi, I cannot read Japanese, but I think I do understand your
diagrams.  You are giving examples for G4, is that correct? I now see why
you say there are 6 stroke partitions of G2. If the two edges of G2 are x
and y, then the six partitions are:

xy'
yx'
x'y
y'x
x + y
x' + y'

I have not confirmed any of the other values -- I have to trust you that
the enumeration in your blog post is correct (yielding 490 stroke
partitions).

But I think it is *still *a bad idea to change the data at A131519, until
we can get an explanation from Max Alexeyev, G. C. Greubel, and R. J.
Mathar, of what they were counting.

Is the recurrence given by Alexeyev correct -- is it just that A(3) and
A(4) are wrong? If A(3) and A(4) were corrected, would the recurrence
produce correct values thereafter?

On Fri, Jan 28, 2022 at 2:36 AM <zbi74583_boat at yahoo.co.jp> wrote:

>         I explain  a little about how to compute the number of Partition
> into stroke    See this figure on my blog, in the manuscript    on 2021/6/27
>     https://mixi.jp/view_diary.pl?id=1979119073&owner_id=716028
>     For instance, I explain { 6 } and { 5,1 }, they represent the cases of
> the partition into one  stroke which has a path made by 6 edges and the
> partition into two  strokes which has two paths  made by 5 edged and one
> edge    The first figure represents 4 vertexes and two edges between them
> and their names, m, j, x, are upper edges, n, k, y, are downer edges    The
> second figure represents a path made by connected six edges, m, j, x, n, k,
> y, names without minus mean right directed edges and names with minus mean
> left directed edges     x 16  means 16 symmetrical partition exist     The
> fifth figure represents two paths  made by five edges, m, j, x, -y,-k, and
> n, plus means union    So, the number of Partition into stroke = 16 + 16 +
> 16 + 16 + 16 + .... + 4 + 16 + 8 + 8 + 2 = 490    > Number of Partition
> into stroke of G_2         j    = right directed e_1, if it has minus then
> it means left         o=o     directed e_1         k    = right directed
> f_1, if it has minus then it means left                directed f_1
> Partitions are ....        j-k, k-j, -jk, -kj, j + k, -j + -k    So, number
> is 6, it is not 2    I give again the definition of Partition into stroke
> of directed graph    It is the partition which satisfies the following two
> conditions    1. The graph is union of the edges which are members of edge
> disjoint directed path s on it
>         ex        x  y  z        o->o->o        xyz is a partition, xy +
> yz is not a partition, because, xy U yz = xyz    Allan    Ask me any thing
> that you don't understand, I will continue  to explain until you understand
> it    I feel the idea is difficult for people who don't use Kanji, I asked
> to confirm to several mathematicians but  they all don't understand it
> If P is a set of paths that satisfies the following three conditions then P
> is a "  Partition of G into stroke  "    1. For all p_i ( p_i el P => p_i <
> G )    2, For all g_j ( g_j el G => ( Exist p_i g_j < p_i )    3. For all
> i,j ( p_i el P and p_j el p => p_i U p_j )
>
>
>     Yasutoshi
>      ----- Original Message -----
>  From: Allan Wechsler <acwacw at gmail.com>
>  To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>  Date: 2022/1/22, Sat 15:03
>  Subject: [seqfan] Re: Partition into Stroke
>
> Yasutoshi, I urge caution.
>
> The definitions at A131519 seem a little confusing and vague to me. For
> example, I am guessing that you do *not* consider v1v2 + v3v2 to be a path,
> even though it satisfies all the constraints that are stated. Other
> constraints, not stated in A131519, are needed to make "path" correspond to
> the usual definition of path in graph theory.
>
> Because the definitions might be confusing, I suspect that Max Alexeyev,
> who is a smart guy, interpreted them to mean something other than what you
> intended. That means that the data listed in A131519 is counting
> *something*,
> but perhaps not the same thing you *wanted* to count.
>
> If that's the case, then what needs to happen is: (1) A131519 should be
> clarified so that it clearly explains what Max was counting. (2) Introduce
> a new sequence with your new data, clearly explaining what *it* is counting
> (and explaining the difference from A131519). (3) Somebody will then step
> up and provide more data, hopefully.
>
> But A131519 is old enough now that there could easily be papers referring
> to it in the mathematical literature, or on websites. So it is probably not
> a good idea to alter the data at this late date.
>
> I think it would help *me *if you could explicitly enumerate the 6 "stroke
> partitions" of G2. I cannot find six of *anything* in G2. G2 only has two
> edges, right?
>
> Allan
>
> On Fri, Jan 21, 2022 at 2:12 AM <zbi74583_boat at yahoo.co.jp> wrote:
>
> > Hi  Seqfans    I abstracted the idea of "Stroke" which is used in writing
> > Kanji.    For instance, when "木" is written, which means tree, these four
> > strokes are used. See " How to write a" 's page
> >    https://kakijun.jp/page/0461200.html
> >    My definition of " Partition into stroke " which is an abstraction of
> > Kanji 's stroke is the following
> >    Given an undirected graph G=(V,E), its partition into strokes is a
> > collection of directed edge-disjoint paths (viewed as sets of directed
> > edges) on V such that (i) union of any two paths is not a path; (ii)
> union
> > of corresponding undirected paths is E.
> >
> >    The other description of the definition is the following
> >    A "stroke" is defined as follows. If the following conditions are
> > satisfied then the partition to directed paths on a directed graph is
> > called "a partition to strokes on a directed graph". And all directed
> paths
> > in the partition are called "strokes". C.1. Two different directed paths
> in
> > a partition do not have the same edges. C.2. A union of two different
> paths
> > in a partition does not become a directed path. In other word, a "stroke"
> > is a locally maximal path on a directed graph.
> >    Recently  I recomputed the terms of A131519 and I have found it is
> > fault
> >    The correct one is    1, 6, 58, 490, ....    So  I am going to rewrite
> > it  but I must confirm  it    Could anyone confirm  it and compute more
> > terms ?    If the definition is difficult then feel free to ask anything
> >
> >
> >    Yasutoshi
> >
> > --
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> >
>
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```