A363893 Number of weakly connected components of an addsub configuration graph with respect to integers mod n over a path with two vertices.

1, 2, 1, 4, 2, 3, 1, 5, 4, 4, 2, 6, 3, 11, 1, 11, 5, 6, 4, 12, 4, 7, 2, 13, 6, 14, 3, 10, 11, 25, 1, 29, 11, 18, 5, 12, 6, 21, 4, 25, 12, 34, 4, 32, 7, 13, 2, 17, 13, 48, 6, 16, 14, 25, 3, 47, 10, 16, 11, 18, 25, 87, 1, 95, 29, 18, 11, 32, 18, 19, 5

Offset: 2

Author

Patrick G. Cesarz, Eugene Fiorini, Charles Gong, Kyle A. Kelley, Philip Thomas, and Andrew Woldar, Jun 26 2023

Keywords

• nonn

Comments

The addsub game is played on a path with two vertices {u,v}. We define a configuration of the integers mod n on {u,v} by assigning weights wt(u) and wt(v).

An addsub move from u to v is a reassignment of weights given by wt(u) -> wt(u) - wt(v) (mod n) and wt(v) -> wt(u) + wt(v) (mod n). An addsub move from v to u is defined analogously.

The addsub configuration graph with respect to the integers mod n over {u,v} is the directed graph in which each node corresponds to a configuration (wt(u),wt(v)) and a directed edge from a configuration to the resulting configuration is attainable via a single addsub move.

Examples

			For n=3, the (u,v) sequence of addsub moves forms the directed cycle (0,1)->(2,1)->(1,0)->(1,1)->(0,2)->(1,2)->(2,0)->(2,2)->(0,1). The (v,u) sequence of addsub moves forms the directed cycle (0,1)->(1,1)->(2,0)->(2,1)->(0,2)->(2,2)->(1,0)->(1,2)->(0,1). These two directed cycles form one weakly connected component. The isolated vertex (0,0) is a loop and forms the second weakly connected component. Therefore, a(3)=2.
		

References

• E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

• E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, Characterizing Winning Positions in the Impartial Two-player Pebbling Game on Complete Graphs, Journal of Integer Sequences, 24(6) (2021).
• E. Fiorini, M. Lind, and A. Woldar, On Properties of Pebble Assignment Graphs, Graphs and Combinatorics, 38(2) (2022), 45.
• E. Fiorini, G. Johnston, M. Lind, A. Woldar, and T. W. H. Wong, Cycles and Girth in Pebble Assignment Graphs, Graphs and Combinatorics, 38(5) (2022), 154.

Crossrefs

Cf. A340631, A346197, A346401.

Programs

Mathematica

Upto=25;
Table[
  VertexSet:={};
  EdgeSet:={};
  (* Compute configuration graph for integers mod n *)
  Do[
    Do[AppendTo[VertexSet,{i,j}];
      AppendTo[EdgeSet,{i,j}\[DirectedEdge]{Mod[i-j,n],Mod[i+j,n]}];
      AppendTo[EdgeSet,{i,j}\[DirectedEdge]{Mod[j+i,n],Mod[j-i,n]}],
      {j,0,n-1}],
    {i,0,n-1}];
  (* Print n-th term *)
  Length[WeaklyConnectedComponents[Graph[VertexSet,EdgeSet]]],
  {n,2,Upto}]