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A241151 Number of distinct degrees in the partition graph G(n) defined at A241150.

%I A241151 #13 Jul 14 2022 17:25:16
%S A241151 1,2,3,3,4,5,5,5,6,7,7,7,7,8,9,9,9,9,9,10,11,11,11,11,11,11,12,13,13,
%T A241151 13,13,13,13,13,14,15,15,15,15,15,15,15,15,16,17,17,17,17,17,17,17,17,
%U A241151 17,18,19,19,19,19,19
%N A241151 Number of distinct degrees in the partition graph G(n) defined at A241150.
%C A241151 a(n) = number of numbers in row n of the array at A241150, counting the top row as row 2.
%C A241151 Conjecture: partial sums of A097806. - _Sean A. Irvine_, Jul 14 2022
%e A241151 (See the Example section of A241150.)
%t A241151     z = 25; spawn[part_] := Map[Reverse[Sort[Flatten[ReplacePart[part, {# - 1, 1}, Position[part, #, 1, 1][[1]][[1]]]]]] &, DeleteCases[DeleteDuplicates[part], 1]];
%t A241151      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 A241151      Column[m] (* A241150 as an array *)
%t A241151      Flatten[m] (* A241150 as a sequence *)
%t A241151      Table[Length[m[[n]]], {n, 1, z}] (* A241151 *)
%t A241151      Table[Max[m[[n]]], {n, 1, z}] (* A241152 *)
%t A241151      Table[Last[m[[n]]], {n, 1, z}] (* A241153 *)
%t A241151      (* Next, show the graph G(k) *)
%t A241151      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 A241151 Cf. A241150, A241152, A241153.
%K A241151 nonn,more
%O A241151 2,2
%A A241151 _Clark Kimberling_ and _Peter J. C. Moses_, Apr 17 2014