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

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

Author

Clark Kimberling and Peter J. C. Moses, Apr 17 2014

Keywords

• nonn
• easy

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.

Examples

			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.)
		

Crossrefs

Cf. A241150, A241151, A241152.

Programs

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 *)