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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. 181
 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: A089178 A187489 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 September 20 20:02 EDT 2020. Contains 337265 sequences. (Running on oeis4.)