cp's OEIS Frontend

A089243 Number of partitions into strokes of the star graph with n edges on the plane, up to rotations and reflections around the center node.

%I A089243 #33 May 14 2023 20:47:32
%S A089243 1,2,3,4,9,22,61,200,689,3054,12110,61132,274264,1515134,7498195,
%T A089243 44301928,238206692,1490114770,8605537805,56612534420,348083793872,
%U A089243 2396294898646,15577794980189,111781094032984,763986810923430,5695585712379834
%N A089243 Number of partitions into strokes of the star graph with n edges on the plane, up to rotations and reflections around the center node.
%C A089243 A "stroke" is defined as follows. If the following conditions are satisfied then the partition to directed paths on a directed graph is called "a partition to strokes on a directed graph", and all directed paths in the partition are called "strokes". C.1. Two different directed paths in a partition do not have the same edges. C.2. A union of two different paths in a partition does not become a directed path. In other words, a "stroke" is a locally maximal path on a directed graph.
%C A089243 This sequence has its origin in the strokes made when writing Japanese Kanji.
%C A089243 The value a(1) is ambiguous as it depends on the definition of the star graph with n = 1 edge. If one of the edge endpoints is labeled as the star center, then we have the current value a(1) = 2. However, if the center is not distinguished, then a(1) would be 1. - _Max Alekseyev_, May 04 2023
%H A089243 Christian Sievers, <a href="/A089243/b089243.txt">Table of n, a(n) for n = 0..728</a>
%e A089243 For n = 3, call the center node "0" and the terminal nodes "1", "2", "3".
%e A089243 Four partitions exist as follows:
%e A089243   {1->0->2, 0->3}
%e A089243   {1->0->2, 3->0}
%e A089243   {1->0, 2->0, 3->0}
%e A089243   {0->1, 0->2, 0->3}.
%e A089243 So a(3) = 4.
%o A089243 (PARI) p(n,t,o)=o*sum(k=0,(n-1)/2,n!/(k!*(n-2*k)!)*t^k)+if(n%2==0, n!/(n/2)!*t^(n/2));
%o A089243 a(n)=if(n==0,1,(sumdiv(n,d,eulerphi(n/d)*p(d,n/d,2)) + if(n%2,2*n*p((n-1)/2,2,1),n/2*p(n/2,2,2)+n*p(n/2-1,2,2)+n*p(n/2-1,2,1)))/(2*n)) \\ _Christian Sievers_, May 14 2023
%Y A089243 Cf. A131518, A131520, A354228, A131709, A357857, A357895.
%K A089243 nonn,walk
%O A089243 0,2
%A A089243 _Yasutoshi Kohmoto_
%E A089243 Edited, terms a(0)-a(1) and a(6) corrected, a(7)-a(13) added by _Max Alekseyev_, Oct 20 2022
%E A089243 More terms from _Christian Sievers_, May 14 2023