%I #64 May 12 2020 10:57:17

%S 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,

%T 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,

%U 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

%N 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.

%C Mirror of triangle A135010.

%H Robert Price, <a href="/A138121/b138121.txt">Table of n, a(n) for n = 1..16851, 25 rows</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">A section model of partitions (2D and 3D)</a> [From _Omar E. Pol_, Sep 07 2008]

%H Robert Price, <a href="/A138121/a138121.txt">Mathematica Program to Generate Diagram</a>

%e Triangle begins:

%e [1];

%e [2],[1];

%e [3],[1],[1];

%e [4],[2,2],[1],[1],[1];

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

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

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

%e ...

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

%e ---------------------------------------------------------

%e Partitions A194805 Table 1.0

%e . of 7 p(n) A194551 A135010

%e ---------------------------------------------------------

%e 7 15 7 7 . . . . . .

%e 4+3 4 4 . . . 3 . .

%e 5+2 5 5 . . . . 2 .

%e 3+2+2 3 3 . . 2 . 2 .

%e 6+1 11 6 1 6 . . . . . 1

%e 3+3+1 3 1 3 . . 3 . . 1

%e 4+2+1 4 1 4 . . . 2 . 1

%e 2+2+2+1 2 1 2 . 2 . 2 . 1

%e 5+1+1 7 1 5 5 . . . . 1 1

%e 3+2+1+1 1 3 3 . . 2 . 1 1

%e 4+1+1+1 5 4 1 4 . . . 1 1 1

%e 2+2+1+1+1 2 1 2 . 2 . 1 1 1

%e 3+1+1+1+1 3 1 3 3 . . 1 1 1 1

%e 2+1+1+1+1+1 2 2 1 2 . 1 1 1 1 1

%e 1+1+1+1+1+1+1 1 1 1 1 1 1 1 1 1

%e . 1 ---------------

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

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

%e . A182713 -------> 1 0 1 2 2

%e . A182714 -------> 1 0 1 1

%e . 1 0 1

%e . A141285 A182703 1 0

%e . A182730 A182731 1

%e ---------------------------------------------------------

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

%e ---------------------------------------------------------

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

%e ---------------------------------------------------------

%e .

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

%e . . . . . 1 . . . .

%e . . . . 2 1 . . . .

%e . . 3 . . 1 2 . . .

%e . Table 2.0 . . 2 2 1 . . 3 . Table 2.1

%e . . . . . 1 2 2 . .

%e . 1 . . . .

%e .

%e . A182982 A182742 A194803 A182983 A182743

%e . A182992 A182994 A194804 A182993 A182995

%e ---------------------------------------------------------

%e .

%e From _Omar E. Pol_, Sep 03 2013: (Start)

%e 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.

%e Illustration of initial terms:

%e ---------------------------------------

%e n j Diagram Parts

%e ---------------------------------------

%e . _

%e 1 1 |_| 1;

%e . _ _

%e 2 1 |_ | 2,

%e 2 2 |_| . 1;

%e . _ _ _

%e 3 1 |_ _ | 3,

%e 3 2 | | . 1,

%e 3 3 |_| . . 1;

%e . _ _ _ _

%e 4 1 |_ _ | 4,

%e 4 2 |_ _|_ | 2, 2,

%e 4 3 | | . 1,

%e 4 4 | | . . 1,

%e 4 5 |_| . . . 1;

%e . _ _ _ _ _

%e 5 1 |_ _ _ | 5,

%e 5 2 |_ _ _|_ | 3, 2,

%e 5 3 | | . 1,

%e 5 4 | | . . 1,

%e 5 5 | | . . 1,

%e 5 6 | | . . . 1,

%e 5 7 |_| . . . . 1;

%e . _ _ _ _ _ _

%e 6 1 |_ _ _ | 6,

%e 6 2 |_ _ _|_ | 3, 3,

%e 6 3 |_ _ | | 4, 2,

%e 6 4 |_ _|_ _|_ | 2, 2, 2,

%e 6 5 | | . 1,

%e 6 6 | | . . 1,

%e 6 7 | | . . 1,

%e 6 8 | | . . . 1,

%e 6 9 | | . . . 1,

%e 6 10 | | . . . . 1,

%e 6 11 |_| . . . . . 1;

%e ...

%e (End)

%t 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 *)

%t 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 *)

%Y Row n has length A138137(n).

%Y Rows sums give A138879.

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

%K nonn,tabf,less

%O 1,2

%A _Omar E. Pol_, Mar 21 2008