A241152 -id:A241152 - cp's OEIS frontend

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

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

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.

			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.

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

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

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 *)

A241151 Number of distinct degrees in the partition graph G(n) defined at A241150.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19

Offset: 2

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

a(n) = number of numbers in row n of the array at A241150, counting the top row as row 2.

Conjecture: partial sums of A097806. - Sean A. Irvine, Jul 14 2022

			(See the Example section of A241150.)

Cf. A241150, A241152, A241153.

    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 *)

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

Original entry on oeis.org

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

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

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 2
1 2 5
1 2 5 10
1 2 5 10 20
1 2 5 10 20 36 ...
    Deleting column 1 leaves
2
2 5
2 5 10
2 5 10 20
2 5 10 20 36... ,
    in which row n is identical to the first n+1 terms of A000712.

			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.)

Cf. A241150, A241151, A241152.

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 *)

cp's OEIS Frontend

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.

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A241151 Number of distinct degrees in the partition graph G(n) defined at A241150.

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Mathematica

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

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Programs

Mathematica