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A026791 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.

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