stroke sequence

[y.kohmoto]

    Hello, seqfans.

    Neil said he already add  a description about my stroke sequence on
reference line of A002620.
    Does it mean he has understood the idea "stroke"?
    But Edwin doesn't seem to have understood it.
    I should explain more about it.

        ----------

    ex.1 a cycle graph of 4 nodes
         ._._.
         |_._|
         names of vertices
         v_00, v_01
         v_10, v_11

    [partition to strokes]
    00->01->11->10->00
    00->01->11->10, 00->10
    00->01->11, 00-.10->11
    00->01, 11->01, 00->10, 11->10
                   where 00, 01 are abbreviations of v_00, v_01
    number of partitions = 4
    maximal number of strokes = 4 , see fourth case

    [classification of di-graph, or orientations of di-graph]
    00->01->11->10->00
    00->01->11->10, 00->10
    00->01->11, 00-.10->11
    00->01, 11->01, 00->10, 11->10
    this is exactly the same as partition to strokes.
    number of di-graphs = 4


    ex.2 a tree graph of 4 branches
          ._._|_._.
             |
         names of vertices
              v_1
         v_4 v_0 v_2
              v_3

    [partition to strokes]
    1->0->2, 3->-0->4
    1->0->2, 4->0->3
    1->0-.3, 2->0->4
    1->0->2, 3->0, 4->0
    1->0->2, 0->3, 0->4
    1->0->3, 2->0, 4->0
    1->0->3, 0->2. 0->4
    1->0, 2->0, 3->0, 4->0
    0->1, 0->2, 0->3, 0->4
    number of partitions = 9
    maximal number of strokes = 4 , see  ninth case

    [classification of di-graph]
    1->0, 2->0, 3->0, 4->0
    0->1, 2->0, 3->0, 4->0
    0->1, 0->2, 3->0, 4->0
    0->1, 2->0, 0->3, 4->0
    1->0, 0->2, 0->3, 0->4
    0->1, 0->2, 0->3, 0->4
    this is different from partition of strokes.
    number of di-graphs = 6

    [Definition of a stroke]
    D.   A locally maximal di-path on di-graph.
    The following condition is necessary.
    C.   Different two strokes don't have the same edges.
    example.
    a "T" graph
         ._|_.
         name of vertices
         v_1
    v_2 v_0 v_3
    If no condition exists, then the following partition to strokes exists.
    1->0->2, 1->0>3
    But if condition D  exists, then it becomes two partitions  as follows.
    1->0->2, 0->3
    1->0->3, 0->2
    Indeed  a Kanji "Hito" is isomorphic to the first set of strokes, and a
Kanji "Ireuru" is isomorphic to the second one.


         ----------

    Edwin Clark wrote :

>Note that an orientation of a complete graph is called a tournament. Seefor
example:
>  http://planetmath.org/?op=getobj&from=objects&name=Tournament
>or
>  http://mathworld.wolfram.com/Tournament.html

    Thank you for teaching me a mathematical term.

>..........there would be binomial(n,2) of them.

    2^binomial(n,2) is the number of all orientations of labeled K_n.

    I think your consideration is too simple, because if you think rotation
or mirror symmetry in n-1 dimension then you will see some of these
orientations of K_n are same.

    Ordinary mathematicians count the number of this classified
orientations.

    What I count is different from it.
    I enumerate the  maximal possibility of  number of strokes on one
partition of K_n.

    Maybe, I should enumerate the number of partition to strokes which is
more than tournament.


    > PS Korewa omoshirois desu.

    Boku mo so omoi masu.

    We are talking in cryptograph?
    A  translation :

    > PS It is interesting.
    I think so too.

    Yasutoshi