cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

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

Keywords

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

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 *)

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

Views

Author

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

Keywords

Comments

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

Examples

			(See the Example section of A241150.)

Crossrefs

Cf. A241150, A241152, A241153.

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 *)

A241152 Maximal number of partitions having the same degree in the partition graph G(n) defined at A241150.

Original entry on oeis.org

2, 2, 3, 3, 4, 6, 8, 10, 13, 17, 22, 32, 43, 57, 77, 94, 119, 144, 178, 209, 274, 364, 465, 597, 746, 935, 1143, 1389, 1674, 2006, 2376, 2803, 3284, 3905, 4853, 6010, 7360, 8988, 10834, 13070, 15565, 18522, 21836, 25713, 30030, 35048, 40575, 46930, 53950
Offset: 2

Views

Author

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

Keywords

Examples

			a(7) counts these 6 partitions:  61, 52, 43, 331, 322, 2221, which all have degree 2 in G(7), as seen by putting k = 7 in the Mathematica program.

Crossrefs

Cf. A241150, A241151, A241153.

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 *)
Showing 1-3 of 3 results.

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