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

%I A241152 #8 Apr 22 2014 22:19:09
%S A241152 2,2,3,3,4,6,8,10,13,17,22,32,43,57,77,94,119,144,178,209,274,364,465,
%T A241152 597,746,935,1143,1389,1674,2006,2376,2803,3284,3905,4853,6010,7360,
%U A241152 8988,10834,13070,15565,18522,21836,25713,30030,35048,40575,46930,53950
%N A241152 Maximal number of partitions having the same degree in the partition graph G(n) defined at A241150.
%e A241152 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.
%t A241152 z = 25; spawn[part_] := Map[Reverse[Sort[Flatten[ReplacePart[part, {# - 1, 1}, Position[part, #, 1, 1][[1]][[1]]]]]] &, DeleteCases[DeleteDuplicates[part], 1]];
%t A241152      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]];
%t A241152      Column[m] (* A241150 as an array *)
%t A241152      Flatten[m] (* A241150 as a sequence *)
%t A241152      Table[Length[m[[n]]], {n, 1, z}] (* A241151 *)
%t A241152      Table[Max[m[[n]]], {n, 1, z}] (* A241152 *)
%t A241152      Table[Last[m[[n]]], {n, 1, z}] (* A241153 *)
%t A241152      (* Next, show the graph G(k) *)
%t A241152      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 *)
%Y A241152 Cf. A241150, A241151, A241153.
%K A241152 nonn,easy
%O A241152 2,1
%A A241152 _Clark Kimberling_ and _Peter J. C. Moses_, Apr 17 2014