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A241153 Number of partitions having the maximal degree in the partition graph G(n) defined at A241150.

%I A241153 #8 Apr 24 2014 17:10:08
%S A241153 2,1,1,2,1,1,2,5,1,1,2,5,10,1,1,2,5,10,20,1,1,2,5,10,20,36,1,1,2,5,10,
%T A241153 20,36,65,1,1,2,5,10,20,36,65,110,1,1,2,5,10,20,36,65,110,185,1,1,2,5,
%U A241153 10,20,36,65,110,185,300
%N A241153 Number of partitions having the maximal degree in the partition graph G(n) defined at A241150.
%C A241153 a(n) = last number in row n of G(n), for n >= 2.  The numbers in this sequence can be formatted as a triangle:
%C A241153     2
%C A241153     1 1 2
%C A241153     1 1 2 5
%C A241153     1 1 2 5 10
%C A241153     1 1 2 5 10 20
%C A241153     1 1 2 5 10 20 36 ...
%C A241153     Deleting column 1 leaves
%C A241153     1 2
%C A241153     1 2 5
%C A241153     1 2 5 10
%C A241153     1 2 5 10 20
%C A241153     1 2 5 10 20 36... ,
%C A241153     in which row n is identical to the first n+1 terms of A000712.
%e A241153 a(9) counts these 5 partitions:  5211, 4311, 42111, 321111, 32211, which all have degree 5, which is maximal for the graph G(9), as seen by putting k = 9 in the Mathematica program.  (See the Example section of A241150.)
%t A241153 z = 25; spawn[part_] := Map[Reverse[Sort[Flatten[ReplacePart[part, {# - 1, 1}, Position[part, #, 1, 1][[1]][[1]]]]]] &, DeleteCases[DeleteDuplicates[part], 1]];
%t A241153      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 A241153      Column[m] (* A241150 as an array *)
%t A241153      Flatten[m] (* A241150 as a sequence *)
%t A241153      Table[Length[m[[n]]], {n, 1, z}] (* A241151 *)
%t A241153      Table[Max[m[[n]]], {n, 1, z}] (* A241152 *)
%t A241153      Table[Last[m[[n]]], {n, 1, z}] (* A241153 *)
%t A241153      (* Next, show the graph G(k) *)
%t A241153      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 A241153 Cf. A241150, A241151, A241152.
%K A241153 nonn,easy
%O A241153 2,1
%A A241153 _Clark Kimberling_ and _Peter J. C. Moses_, Apr 17 2014