A241152 Maximal number of partitions having the same degree in the partition graph G(n) defined at A241150.

2, 2, 3, 3, 4, 6, 8, 10, 13, 17, 22, 32, 43, 57, 77, 94, 119, 144, 178, 209, 274, 364, 465, 597, 746, 935, 1143, 1389, 1674, 2006, 2376, 2803, 3284, 3905, 4853, 6010, 7360, 8988, 10834, 13070, 15565, 18522, 21836, 25713, 30030, 35048, 40575, 46930, 53950

Offset: 2

Author

Clark Kimberling and Peter J. C. Moses, Apr 17 2014

Keywords

• nonn
• easy

Examples

			a(7) counts these 6 partitions:  61, 52, 43, 331, 322, 2221, which all have degree 2 in G(7), as seen by putting k = 7 in the Mathematica program.
		

Crossrefs

Cf. A241150, A241151, A241153.

Programs

Mathematica

z = 25; spawn[part_] := Map[Reverse[Sort[Flatten[ReplacePart[part, {# - 1, 1}, Position[part, #, 1, 1][[1]][[1]]]]]] &, DeleteCases[DeleteDuplicates[part], 1]];
     unspawn[part_] := If[Length[Cases[part, 1]] > 0, Map[ReplacePart[Most[part], Position[Most[part], #, 1, 1][[1]][[1]] -> # + 1] &, DeleteDuplicates[Most[part]]], {}]; m = Map[Last[Transpose[Tally[Map[#[[2]] &, Tally[Flatten[{Map[unspawn, #], Map[spawn, #]}, 2] &[IntegerPartitions[#]]]]]]] &, 1 + Range[z]];
     Column[m] (* A241150 as an array *)
     Flatten[m] (* A241150 as a sequence *)
     Table[Length[m[[n]]], {n, 1, z}] (* A241151 *)
     Table[Max[m[[n]]], {n, 1, z}] (* A241152 *)
     Table[Last[m[[n]]], {n, 1, z}] (* A241153 *)
     (* Next, show the graph G(k) *)
     k = 8; graph = Flatten[Table[part = IntegerPartitions[k][[n]]; Map[FromDigits[part] -> FromDigits[#] &, spawn[part]], {n, 1, PartitionsP[k]}]]; Graph[graph, VertexLabels -> "Name", ImageSize -> 500, ImagePadding -> 20] (* Peter J. C. Moses, Apr 15 2014 *)