# A109094 and A131709

Sat Dec 15 02:21:57 CET 2007

```    Dear Artur
I think that A109094 is interesting
And I want to know the number of all different maximal paths in K_n.
Because it is a part of "Bus route problem" on K_n.

Each maximal path has partitions into rootes of subgraph {K_n - the maximal path} and they give total partitions into "Bus routes" on K_n.
So, the number of partitions into "Bus routes" on K_n is much more than the number of maximal path in K_n.

"Bus route on 1 x n grid" ---> A131709

Once I submitted a sequence which is number of maximal directed path in 1 x n grid graph.
But Neil rejected it.
He seemed to think that it is not so interesting.

A000001 is not interesting at, but A000002 is rather interesting.

Yasutoshi

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

%I A000001
%S A000001 1, 4, 7, 9, 11
%N A000001 Length of maximal path on 1 x n grid.
%C A000001     If 1<n then a(n)=2*n+3
%e A000001      n=4
.__.__.__.__.
|__|__|__|__|
Names of nodes
1 2 3 4 5
a b c d e
e.g.
345edcba123c is a maximal directed path. So,
a(4)=11
%Y A000001 A000002, A049486
%K A000001 none
%O A000001 0,2
%A A000001 Yasutsohi Kohmoto

%I A000002
%S A000002 2, 8, 12, 40
%N A000002 Number of maximal directed path on 1 x n grid.
%e A000002      n=3
.__.__.__.
|__|__|__|
Names of nodes
1 2 3 4
a b c d
Maximal directed paths which start from node 3.
34dcba123c
34dc32ba12
34dc321ab2
34dc321abc
3cd432ba12
3cd4321ab2
3cd4321abc
3cba1234dc
321abc34dc
321abcd43c
Paths from nodes c,b,2 exist.
So, n(3)=4*10=40
%Y A000002 A000001, A089243
%K A000002 none
%O A000002 0,1

%A A000002 Yasutoshi Kohmoto   zbi74583 at boat.zero.ad.jp

%I A000003
%S A000003 1,1,3,63
%N A000003 Number of partitions into "Bus route" of K_n.
%C A000003   1. One and only one route exists on all edges of G
2. Terminals of two different routes don't meet on the same point

%Y A000002 A131709
%K A000002 none
%O A000002 1,3
%A A000002 Yasutoshi Kohmoto   zbi74583 at boat.zero.ad.jp

```