Partition into strokes

[N. J. A. Sloane]

----- Original Message -----

送信者：Max Alekseyev <maxale at gmail.com>
宛先：koh <zbi74583 at boat.zero.ad.jp>
回送先：seqfan at ext.jussieu.fr
件名：Re: RE : Partition into strokes
送信日時：2007年8月23日 

>On 8/22/07, koh <zbi74583 at boat.zero.ad.jp> wrote:
>
>> >     Or, "Partition of a graph G into strokes S_i" must satisfy the following conditions.
>> >
>> >     o Union_{i} S_i = H
>> >     o If not{i=j} -> S_i and S_j don't have the same edge
>> >     o If not{i=j} -> S_i U S_j isn't a dipath
>> >     o For all i S_i is a dipath
>> >          Where H is a digraph on G
>
>[...]
>
>>          "Partition of a graph G into strokes" means "Partition of a digraph H on graph G into strokes".
>
>But what is exactly "a digraph on graph" ?
>
    Does the phrase  "a digraph on graph" mean nothing?
    If so then it is a problem of my English.

    I want to say the following : 

    If all directed edges of a digraph H are replaced by ordinal edges then H becomes graph G.
    I write this relationship between H and G as follows.

         "a digraph H on graph G"

    example : o->o is a digraph on o-o 



>>          See the four conditions.
>>
>>     2) n=3
>>          o-o-o   names of vertices 1-2-3
>>
>>          Partitions into strokes :
>>          1->2->3
>>          3->2->1
>>          1->2, 3->2
>>          2->1, 2->3
>>          So, a(3)=4
>
>I'm confused. All listed partitions represent partitions of
>*different* digraphs.

    You are right.
    I calculated the number of partitions into strokes of all different digraphs on the G_n.

>Say, the first partition is of the digraph ( 1 -> 2 -> 3 ) while the
>last partition is of the digraph ( 1 <- 2 -> 3 ). Clearly, these
>digraphs are different.
>So, what is H in this case and what exactly the equality "Union_{i}
>S_i = H" from your definition means?
>
    I am sorry.
    My description is ambiguous.
    
    I should have written as H_i for each digraphs on G_n.
    On this case, four digraphs H_i on G exist and each digraphs have one partition into strokes.

    example : last partition 
              Union_{i|1<=i<=2} S_i = S_1 U S_2 = 2->1 U 2->3 = 1<-2->3 = H_4

>Regards,
>Max
>



    Yasutoshi