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A241151
Number of distinct degrees in the partition graph G(n) defined at A241150.
4
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, 13, 13, 13, 13, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19
OFFSET
2,2
COMMENTS
a(n) = number of numbers in row n of the array at A241150, counting the top row as row 2.
Conjecture: partial sums of A097806. - Sean A. Irvine, Jul 14 2022
EXAMPLE
(See the Example section of A241150.)
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,more
AUTHOR
STATUS
approved