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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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 is 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: A236468 A116685 A268190 * A051135 A260258 A283196

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

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Last modified April 25 00:46 EDT 2017. Contains 285346 sequences.