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A138121 Triangle read by rows in which row n lists the partitions of n that do not contain 1 as a part in juxtaposed reverse-lexicographical order followed by A000041(n-1) 1's. 196
1, 2, 1, 3, 1, 1, 4, 2, 2, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 6, 3, 3, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 4, 3, 5, 2, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 4, 5, 3, 6, 2, 3, 3, 2, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 5, 4, 6, 3, 3, 3, 3, 7, 2, 4, 3, 2, 5, 2, 2, 3, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Mirror of triangle A135010.

LINKS

Robert Price, Table of n, a(n) for n = 1..16851, 25 rows

Omar E. Pol, A section model of partitions (2D and 3D) [From Omar E. Pol, Sep 07 2008]

Robert Price, Mathematica Program to Generate Diagram

EXAMPLE

Triangle begins:

[1];

[2],[1];

[3],[1],[1];

[4],[2,2],[1],[1],[1];

[5],[3,2],[1],[1],[1],[1],[1];

[6],[3,3],[4,2],[2,2,2],[1],[1],[1],[1],[1],[1],[1];

[7],[4,3],[5,2],[3,2,2],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1];

...

The illustration of the three views of the section model of partitions (version "tree" with seven sections) shows the connection between several sequences.

---------------------------------------------------------

Partitions A194805 Table 1.0

. of 7 p(n) A194551 A135010

---------------------------------------------------------

7 15 7 7 . . . . . .

4+3 4 4 . . . 3 . .

5+2 5 5 . . . . 2 .

3+2+2 3 3 . . 2 . 2 .

6+1 11 6 1 6 . . . . . 1

3+3+1 3 1 3 . . 3 . . 1

4+2+1 4 1 4 . . . 2 . 1

2+2+2+1 2 1 2 . 2 . 2 . 1

5+1+1 7 1 5 5 . . . . 1 1

3+2+1+1 1 3 3 . . 2 . 1 1

4+1+1+1 5 4 1 4 . . . 1 1 1

2+2+1+1+1 2 1 2 . 2 . 1 1 1

3+1+1+1+1 3 1 3 3 . . 1 1 1 1

2+1+1+1+1+1 2 2 1 2 . 1 1 1 1 1

1+1+1+1+1+1+1 1 1 1 1 1 1 1 1 1

. 1 ---------------

. *<------- A000041 -------> 1 1 2 3 5 7 11

. A182712 -------> 1 0 2 1 4 3

. A182713 -------> 1 0 1 2 2

. A182714 -------> 1 0 1 1

. 1 0 1

. A141285 A182703 1 0

. A182730 A182731 1

---------------------------------------------------------

. A138137 --> 1 2 3 6 9 15..

---------------------------------------------------------

. A182746 <--- 4 . 2 1 0 1 2 . 4 ---> A182747

---------------------------------------------------------

.

. A182732 <--- 6 3 4 2 1 3 5 4 7 ---> A182733

. . . . . 1 . . . .

. . . . 2 1 . . . .

. . 3 . . 1 2 . . .

. Table 2.0 . . 2 2 1 . . 3 . Table 2.1

. . . . . 1 2 2 . .

. 1 . . . .

.

. A182982 A182742 A194803 A182983 A182743

. A182992 A182994 A194804 A182993 A182995

---------------------------------------------------------

.

From Omar E. Pol, Sep 03 2013: (Start)

Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6. Note that before the dissection the set of partitions was in the ordering mentioned in A026792. More generally, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.

Illustration of initial terms:

---------------------------------------

n j Diagram Parts

---------------------------------------

. _

1 1 |_| 1;

. _ _

2 1 |_ | 2,

2 2 |_| . 1;

. _ _ _

3 1 |_ _ | 3,

3 2 | | . 1,

3 3 |_| . . 1;

. _ _ _ _

4 1 |_ _ | 4,

4 2 |_ _|_ | 2, 2,

4 3 | | . 1,

4 4 | | . . 1,

4 5 |_| . . . 1;

. _ _ _ _ _

5 1 |_ _ _ | 5,

5 2 |_ _ _|_ | 3, 2,

5 3 | | . 1,

5 4 | | . . 1,

5 5 | | . . 1,

5 6 | | . . . 1,

5 7 |_| . . . . 1;

. _ _ _ _ _ _

6 1 |_ _ _ | 6,

6 2 |_ _ _|_ | 3, 3,

6 3 |_ _ | | 4, 2,

6 4 |_ _|_ _|_ | 2, 2, 2,

6 5 | | . 1,

6 6 | | . . 1,

6 7 | | . . 1,

6 8 | | . . . 1,

6 9 | | . . . 1,

6 10 | | . . . . 1,

6 11 |_| . . . . . 1;

...

(End)

MATHEMATICA

less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[Array[1 &, {PartitionsP[n - 1]}], Sort[Reverse /@ Select[IntegerPartitions[n], FreeQ[#, 1] &], less]] // Flatten // Reverse; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 15 2013 *)

Table[Reverse/@Reverse@DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x]!=1]], x_ /; x==0, 2]~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 1, 9}] // Flatten (* Robert Price, May 11 2020 *)

CROSSREFS

Row n has length A138137(n).

Rows sums give A138879.

Cf. A000041, A135010, A138879, A138880, A141285, A182703, A194812, A206437, A211009.

Sequence in context: A187489 A355145 A116599 * A138151 A207378 A166556

Adjacent sequences: A138118 A138119 A138120 * A138122 A138123 A138124

KEYWORD

nonn,tabf,less

AUTHOR

Omar E. Pol, Mar 21 2008

STATUS

approved

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Last modified December 9 23:05 EST 2022. Contains 358710 sequences. (Running on oeis4.)