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%I #79 Sep 22 2023 05:17:07
%S 1,1,1,2,1,1,1,1,2,3,1,1,1,1,1,1,2,1,3,2,2,4,1,1,1,1,1,1,1,1,2,1,1,3,
%T 1,2,2,1,4,2,3,5,1,1,1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,2,2,1,1,4,1,2,3,1,
%U 5,2,2,2,2,4,3,3,6,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,1,1,1,2,2,1,1,1,4,1,1,2,3,1,1,5
%N Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.
%C Differs from A080576 in a(18): Here, (...,1+3,2+2,4), there (...,2+2,1+3,4).
%C The representation of the partitions (for fixed n) is as (weakly) increasing lists of parts, the order between individual partitions (for the same n) is lexicographic (see example). - _Joerg Arndt_, Sep 03 2013
%C The equivalent sequence for compositions (ordered partitions) is A228369. - _Omar E. Pol_, Oct 19 2019
%H Alois P. Heinz, <a href="/A026791/b026791.txt">Rows n = 1..19, flattened</a>
%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e First six rows are:
%e [[1]];
%e [[1, 1], [2]];
%e [[1, 1, 1], [1, 2], [3]];
%e [[1, 1, 1, 1], [1, 1, 2], [1, 3], [2, 2], [4]];
%e [[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 4], [2, 3], [5]];
%e [[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2], [1, 1, 1, 3], [1, 1, 2, 2], [1, 1, 4], [1, 2, 3], [1, 5], [2, 2, 2], [2, 4], [3, 3], [6]];
%e ...
%e From _Omar E. Pol_, Sep 03 2013: (Start)
%e Illustration of initial terms:
%e ----------------------------------
%e . Ordered
%e n j Diagram partition j
%e ----------------------------------
%e . _
%e 1 1 |_| 1;
%e . _ _
%e 2 1 | |_| 1, 1,
%e 2 2 |_ _| 2;
%e . _ _ _
%e 3 1 | | |_| 1, 1, 1,
%e 3 2 | |_ _| 1, 2,
%e 3 3 |_ _ _| 3;
%e . _ _ _ _
%e 4 1 | | | |_| 1, 1, 1, 1,
%e 4 2 | | |_ _| 1, 1, 2,
%e 4 3 | |_ _ _| 1, 3,
%e 4 4 | |_ _| 2, 2,
%e 4 5 |_ _ _ _| 4;
%e ...
%e (End)
%p T:= proc(n) local b, ll;
%p b:= proc(n,l)
%p if n=0 then ll:= ll, l[]
%p else seq(b(n-i, [l[], i]), i=`if`(l=[],1,l[-1])..n)
%p fi
%p end;
%p ll:= NULL; b(n, []); ll
%p end:
%p seq(T(n), n=1..8); # _Alois P. Heinz_, Jul 16 2011
%t T[n0_] := Module[{b, ll}, b[n_, l_] := If[n == 0, ll = Join[ll, l], Table[ b[n - i, Append[l, i]], {i, If[l == {}, 1, l[[-1]]], n}]]; ll = {}; b[n0, {}]; ll]; Table[T[n], {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Aug 05 2015, after _Alois P. Heinz_ *)
%t Table[DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions[n]], x_ /; x == 0, 2], {n, 7}] // Flatten (* _Robert Price_, May 18 2020 *)
%o (Python)
%o t = [[[]]]
%o for n in range(1, 10):
%o p = []
%o for minp in range(1, n):
%o p += [[minp] + pp for pp in t[n-minp] if min(pp) >= minp]
%o t.append(p + [[n]])
%o print(t)
%o # _Andrey Zabolotskiy_, Oct 18 2019
%Y Row lengths are given in A006128.
%Y Partition lengths are in A193173.
%Y Other partition orderings: A026792, A036037, A080577, A125106, A139100, A181087, A181317, A182937, A228100, A240837, A242628.
%Y Row lengths are A000041.
%Y Partition sums are A036042.
%Y Partition minima are A196931.
%Y Partition maxima are A194546.
%Y The reflected version is A211992.
%Y The length-sensitive version (sum/length/lex) is A036036.
%Y The colexicographic version (sum/colex) is A080576.
%Y The version for non-reversed partitions is A193073.
%Y Compositions under the same ordering (sum/lex) are A228369.
%Y The reverse-lexicographic version (sum/revlex) is A228531.
%Y The Heinz numbers of these partitions are A334437.
%Y Cf. A049085, A103921, A112798, A115623, A129129, A331581, A334435, A334439, A334442.
%K nonn,tabf
%O 1,4
%A _Clark Kimberling_
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