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

2,1

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.

Links

Table of n, a(n) for n=2..65.

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

Cf. A241150, A241151, A241152.

Sequence in context: A326371 A226304 A101022 * A213852 A051064 A280509

Adjacent sequences: A241150 A241151 A241152 * A241154 A241155 A241156

Keyword

nonn,easy

Author

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

Status

approved