A241150 Irregular triangle read by rows T(n,k) = number of partitions of degree k in the partition graph G(n), for n >= 2; G(n) is defined in Comments.

2, 2, 1, 3, 1, 1, 2, 3, 2, 4, 2, 4, 1, 2, 6, 5, 1, 1, 4, 5, 8, 3, 2, 3, 8, 10, 4, 5, 4, 10, 13, 5, 9, 1, 2, 13, 17, 8, 14, 1, 1, 6, 12, 22, 10, 22, 3, 2, 2, 19, 27, 11, 32, 5, 5, 4, 21, 33, 15, 43, 9, 10, 4, 20, 44, 21, 57, 10, 19, 1, 5, 28, 50, 20, 77, 20

Offset

1,1

Comments

The partition graph G(n) of n has the partitions of n as nodes, and nodes p and q have an edge if one of them can be obtained from the other by a substitution x -> x-1,1 for some part x. G(n) is nonplanar for n >= 8. Column 1: divisors of n, A000005(n), for n >= 2. A000041(n) = sum of numbers in row n, for n >= 2 (counting the top row as row 2). Number of numbers in row n (i.e., maximal degree in G(n)): A241151(n), n >= 2. Last term in row n (the number of partitions having maximal degree): A241153(n), n >= 2. Maximal number in row n: A241152(n), n >= 2. Let u(n,k) be the array at A029205 (where n >= 0, k=0..n). Then u(n,k) is the number of edges in G(n+2) between partitions of n+2 that having length k+1 and those having length k+2.

Links

Clark Kimberling, Table of n, a(n) for n = 1..500

Example

The first 12 rows:
  2
  2 ... 1
  3 ... 1 ... 1
  2 ... 3 ... 2
  4 ... 2 ... 4 ... 1
  2 ... 6 ... 5 ... 1 ... 1
  4 ... 5 ... 8 ... 3 ... 2
  3 ... 8 ... 10 .. 4 ... 5
  4 ... 10 .. 13 .. 5 ... 9 ... 1
  2 ... 13 .. 17 .. 8 ... 14 .. 1 ... 1
  6 ... 12 .. 22 .. 10 .. 22 .. 3 ... 2
  2 ... 19 .. 27 .. 11 .. 32 .. 5 ... 5
The graph can be represented by these transformations:
6 -> 51, 51 -> 411, 42 -> 321, 42 -> 411, 411 -> 3111, 33 -> 321, 321 -> 2211, 321 -> 3111, 3111 -> 21111, 222 -> 2211, 2211 -> 21111, 21111 -> 111111.  These 4 partitions p have degree 1 (i.e., number of arrows to or from p): 6, 33, 222, 111111; these 2 have degree 2: 51, 42; these 4 have degree 3: 411, 3111, 2211, 21111; the remaining partition, 321, has degree 4. So, row 6 of the array is 4 2 4 1.

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. A241151, A241152, A241153, A029205, A000041.

Sequence in context: A116685 A355522 A268190 * A051135 A325541 A260258

Adjacent sequences: A241147 A241148 A241149 * A241151 A241152 A241153

Keyword

nonn,easy,tabf

Author

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

Status

approved