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

2,1

Links

Table of n, a(n) for n=2..50.

Example

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.

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 *)

Crossrefs

Cf. A241150, A241151, A241153.

Sequence in context: A146930 A333524 A319728 * A367843 A164529 A153906

Adjacent sequences: A241149 A241150 A241151 * A241153 A241154 A241155

Keyword

nonn,easy

Author

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

Status

approved