login
A241152
Maximal number of partitions having the same degree in the partition graph G(n) defined at A241150.
4
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved