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A241153
Number of partitions having the maximal degree in the partition graph G(n) defined at A241150.
4
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, 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, 10, 20, 36, 65, 110, 185, 300
OFFSET
2,1
COMMENTS
a(n) = last number in row n of G(n), for n >= 2. The numbers in this sequence can be formatted as a triangle:
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 ...
Deleting column 1 leaves
1 2
1 2 5
1 2 5 10
1 2 5 10 20
1 2 5 10 20 36... ,
in which row n is identical to the first n+1 terms of A000712.
EXAMPLE
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.)
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