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<DIV><FONT face="MS UI Gothic" size=2> Hi,
Seqfans</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> I calculated
"</FONT><FONT face="MS UI Gothic" size=2>Length of maximal di-path on grid graph
n x n", but it already existed on OEIS.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> It is
A049486. <BR>
.__.__.__.__.<BR>
|__|__|__|__| <BR>
|__|__|__|__| <BR>
|__|__|__|__|<BR> |__|__|__|__|
<BR> <BR> S_0
: 0, 4, 10, 21, 34, 53, <BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> Then I tried to
calculate "Number of possible maximal di-paths on grid graph n x
n" </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 0 x 0</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> .</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> Name of
node</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 0</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> No di-path exists. So,
a(0)=0</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 1 x
1 </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> </FONT><FONT face="MS UI Gothic"
size=2> .__</FONT><FONT
face="MS UI Gothic" size=2>.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> |__| </FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2>
Names of nodes</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 12</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 34 </FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 12431</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 13421</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 2 di-paths exist. So,
a(1)=2. Where If a di-path is a rotation of another one, it is
not counted.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 2 x 2</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> .__.__.<BR></FONT><FONT
face="MS UI Gothic" size=2> </FONT><FONT
face="MS UI Gothic" size=2>
|__|__|<BR>
|__|__| <BR> <BR>
Names of nodes</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
123<BR>
456<BR> 789<BR></DIV></FONT>
<DIV><FONT face="MS UI Gothic"
size=2>
23698741256<BR>
23698741258<BR>
23698741254</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 23698745214</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
23698541256<BR>
23698521456<BR> <BR>
23658741254<BR>
23658745214<BR>
<BR> 23654785214</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
23654125896<BR>
23654125874</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
25632145896<BR>
25632145874</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 25632147854</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
25698741236<BR> <BR>
25698541236</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
25896541236<BR>
25896321456<BR> 25896321478
</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
25874123698<BR>
25874123654<BR>
<BR>
25478563214<BR>
25478963214</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
25412369874<BR>
25412369856<BR>
25412365896<BR>
25412365874</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"><FONT
size=2><FONT>
21478523698<BR>
21478523654<BR>
21478563254<BR> 21478965236
<BR>
21478963256 <BR>
21478963258<BR>
21478963254<BR></FONT></FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 34 di-paths exist.
So, a(2)=34.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> S_1 : 0, 2,
34</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> I counted
it by hand. Is it correct?</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> I should write a soft
ware which calculates this sequence.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> It is essentially
a problem of counting all branches of a directed tree
graph.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> Because the
union of these di-paths becomes "a tree di-graph" which covers a part of a grid
graph.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> I understand that
this description has a little ambiguousness.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> I want to know the
exact mathematical term.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> How do
mathematicians call the following mapping.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> eg.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> Grid
graph G_1. 1 x 1 :
</DIV>
<DIV><FONT face="MS UI Gothic" size=2> </FONT><FONT face="MS UI Gothic"
size=2> .__</FONT><FONT
face="MS UI Gothic" size=2>.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> |__| </FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2>
Names of nodes</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 12</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> 34
</FONT></DIV></FONT>
<DIV><FONT face="MS UI Gothic" size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> Tree di-graph T_1. :
</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> V={a,b,c,d,e,f,g,h,i}</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
E={ab,bc,cd,de,af,fg,gh,hi}</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> Mapping
F T_1 -> G_1 : </FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> F(ab)=12, F(bc)=24,
F(cd)=43, F(de)=31,</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> F(af)=13, F(fg)=34,
F(gh)=42, F(hi)=21</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> S_0 and S_1 are a
part of the study of a sequence "Number of partitions into strokes of grid graph
n x n" </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> For example, the first
di-path of the case 2 x 2, "23698741256" conducts four partitions
into strokes..</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
23698741256+458</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
23698741256+854</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
23698741256+45+85</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2>
23698741256+54+58 </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"
size=2> Yasutoshi</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2>
</DIV></FONT></BODY></HTML>