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A138121
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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.
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196
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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
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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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.
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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
. 1 0 1
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. . . . . 1 . . . .
. . . . 2 1 . . . .
. . 3 . . 1 2 . . .
. Table 2.0 . . 2 2 1 . . 3 . Table 2.1
. . . . . 1 2 2 . .
. 1 . . . .
.
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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:
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n j Diagram Parts
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. _
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)
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf,less
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AUTHOR
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STATUS
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approved
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