OFFSET
1,2
COMMENTS
Note that the terms in the n-th row are the same as the terms in the n-th row of triangle A280851, but in some rows the terms appear in distinct order. First differs from A280851 at a(28) = T(15,3). - Omar E. Pol, Apr 24 2018
Row n in the triangle is a sequence of A250068(n) symmetric sections, each section consisting of the sizes of the subparts on that level in the symmetric representation of sigma of n - from the top down in the images below or left to right as drawn in A237593. - Hartmut F. W. Hoft, Sep 05 2021
EXAMPLE
Triangle begins (first 15 rows):
[1];
[3];
[2, 2];
[7];
[3, 3];
[11], [1];
[4, 4];
[15];
[5, 3, 5];
[9, 9];
[6, 6];
[23], [5];
[7, 7];
[12, 12];
[8, 7, 8], [1];
...
For n = 12 we have that the 11th row of triangle A237593 is [6, 3, 1, 1, 1, 1, 3, 6] and the 12th row of the same triangle is [7, 2, 2, 1, 1, 2, 2, 7], so the diagram of the symmetric representation of sigma(12) = 28 is constructed as shown below in Figure 1:
. _ _
. | | | |
. | | | |
. | | | |
. | | | |
. | | | |
. _ _ _| | _ _ _| |
. _| _ _| _| _ _ _|
. _| | _| _| |
. | _| | _| _|
. | _ _| | |_ _|
. _ _ _ _ _ _| | 28 _ _ _ _ _ _| | 5
. |_ _ _ _ _ _ _| |_ _ _ _ _ _ _|
. 23
.
. Figure 1. The symmetric Figure 2. After the dissection
. representation of sigma(12) of the symmetric representation
. has only one part which of sigma(12) into layers of
. contains 28 cells, so width 1 we can see two "subparts"
. the 12th row of the that contain 23 and 5 cells
. triangle A237270 is [28]. respectively, so the 12th row of
. this triangle is [23], [5].
.
For n = 15 we have that the 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8], so the diagram of the symmetric representation of sigma(15) = 24 is constructed as shown below in Figure 3:
. _ _
. | | | |
. | | | |
. | | | |
. | | | |
. | | | |
. | | | |
. | | | |
. _ _ _|_| _ _ _|_|
. _ _| | 8 _ _| | 8
. | _| | _ _|
. _| _| _| |_|
. |_ _| 8 |_ _| 1
. | | 7
. _ _ _ _ _ _ _ _| _ _ _ _ _ _ _ _|
. |_ _ _ _ _ _ _ _| |_ _ _ _ _ _ _ _|
. 8 8
.
. Figure 3. The symmetric Figure 4. After the dissection
. representation of sigma(15) of the symmetric representation
. has three parts of size 8 of sigma(15) into layers of
. because every part contains width 1 we can see four "subparts".
. 8 cells, so the 15th row of The first layer has three subparts:
. triangle A237270 is [8, 8, 8]. 8, 7, 8. The second layer has
. only one subpart of size 1, so
. the 15th row of this triangle is
. [8, 7, 8], [1].
.
The smallest even number with 3 levels is 60; its row of subparts is: [119], [37], [6, 6]. The smallest odd number with 3 levels is 315; its row of subparts is: [158, 207, 158], [11, 26, 5, 9, 5, 26, 11], [4, 4]. - Hartmut F. W. Hoft, Sep 05 2021
MATHEMATICA
(* support functions are defined in aA237593 and A262045 *)
subP[level_] := Module[{s=Map[Apply[Plus, #]&, Select[level, First[#]!=0&]]}, If[OddQ[Length[s]], s[[(Length[s]+1)/2]]-=1]; s]
a279391[n_] := Module[{widL=a262045[n], lenL=a237593[n], srs, subs}, srs=Transpose[Map[PadRight[If[widL[[#]]>0, Table[1, widL[[#]]], {0}], Max[widL]]&, Range[Length[lenL]]]]; subs=Map[SplitBy[lenL srs[[#]], #!=0&]&, Range[Max[widL]]]; Flatten[Map[subP, subs]]]
Flatten[Map[a279391, Range[38]]] (* Hartmut F. W. Hoft, Sep 05 2021 *)
CROSSREFS
The length of row n equals A001227(n).
If n is odd the length of row n equals A000005(n).
Row sums give A000203.
For the definition of "subparts" see A279387.
For the triangle of sums of subparts see A279388.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 12 2016
STATUS
approved