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

Author

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

Keywords

• nonn
• easy
• tabf

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A241151, A241152, A241153, A029205, A000041.

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 *)