Conjectured formula for A004081 from Jon Schoenfield

[N. J. A. Sloane]

seems very interesting. - Neil
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Date: Fri, 10 Aug 2007 15:50:33 +0900
Subject: Partition into strokes
From: "koh" <zbi74583 at boat.zero.ad.jp>
To: seqfan at ext.jussieu.fr
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    Hi, Seqfans

    Once I considered about "Strokes" of a graph.

    See the definition in A089243.
         
    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 

    The idea seems to have not existed in Mathematics.
    I think , " at least" , to count number of partitions of a graph into strokes is interesting. 

    I am going to submit some sequences of numbers of partitions of graphs.
    Could anyone verify them?

    They are rather easy graphs, because counting partitions of a graph is    very complicated.

    I don't know number of partitions of Grid graph 2x2 into strokes.
    If anyone knows the exact number then mail it to me. 



    --------------------



    %S A000001 1,2,4,8,16,32,64
    %N A000001 Number of partitions of G_n into "strokes".
                G_n = {V_n, E_n}, V_n = {v_1, v_2,$B!D(B.v_n}, E_n = {v_1v_2, v2_v_3,$B!D(B.v_{n-1}v_n} 
                     
                 Figure of G_5 : o-o-o-o-o 


$B!!(B
    %S A000002 2,6,14,104
    %N A000002 Number of partitions of G_n into "strokes".
                 G_n = {V_n, E_n}, V_n = {v_1, v_2}, E_n = {e_1, e_2, $B!D(B.e_n}, For all {i}  e_i = v_1v_2  
                     
                 Figure of G_2  : o=o    Two edges exist between v_1 and v_2 .   

                                    


    %S A000003 1,6,66
    %N A000003 Number of partitions of G_n into "strokes".
                 G_n = {V_n, E_n}, V_n = {v_1, v_2,$B!D(B.v_n}, E_n = {e_1, e_2, $B!D(B. e_{n-1}, f_1, f_2, $B!D(B. f_{n-1}}, For all {i}  e_i = f_i = v_iv_{i+1}
                          
                 Figure of G_5 : o=o=o=o=o




    %S A000004 2,6,12,22,40,80,168
    %N A000004 Number of partitions of G_n into "strokes".
                 G_n = {V_n, E_n}, V_n = {v_1, v_2,$B!D(B.v_n}, E_n = {v_1v_2, v2_v_3,$B!D(B.v_{n-1}v_n, v_nv_1}  
                     
                 Figure of G_4 : o-o-o-o-o    Two vertices of both sides are the same.



    Yasutoshi