Conjectured formula for A004081 from Jon Schoenfield
N. J. A. Sloane
njas at research.att.com
Fri Aug 10 07:40:48 CEST 2007
seems very interesting. - Neil
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Date: Fri, 10 Aug 2007 15:50:33 +0900
Subject: Partition into strokes
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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,….v_n}, E_n = {v_1v_2, v2_v_3,….v_{n-1}v_n}
Figure of G_5 : o-o-o-o-o
%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, ….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,….v_n}, E_n = {e_1, e_2, …. e_{n-1}, f_1, f_2, …. 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,….v_n}, E_n = {v_1v_2, v2_v_3,….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
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