cp's OEIS Frontend

A026792 List of juxtaposed reverse-lexicographically ordered partitions of the positive integers.

%I A026792 #58 Aug 06 2024 00:02:51
%S A026792 1,2,1,1,3,2,1,1,1,1,4,2,2,3,1,2,1,1,1,1,1,1,5,3,2,4,1,2,2,1,3,1,1,2,
%T A026792 1,1,1,1,1,1,1,1,6,3,3,4,2,2,2,2,5,1,3,2,1,4,1,1,2,2,1,1,3,1,1,1,2,1,
%U A026792 1,1,1,1,1,1,1,1,1,7,4,3,5,2,3,2,2,6,1,3,3,1,4,2,1,2,2,2,1,5,1,1,3,2,1,1,4,1,1,1,2,2,1
%N A026792 List of juxtaposed reverse-lexicographically ordered partitions of the positive integers.
%C A026792 The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is (list-)reversed lexicographic; see examples. [_Joerg Arndt_, Sep 03 2013]
%C A026792 Written as a triangle; row n has length A006128(n); row sums give A066186. Also written as an irregular tetrahedron in which T(n,j,k) is the k-th largest part of the j-th partition of n; the sum of column k in the slice n is A181187(n,k); right border of the slices gives A182715. - _Omar E. Pol_, Mar 25 2012
%C A026792 The equivalent sequence for compositions (ordered partitions) is A228351. - _Omar E. Pol_, Sep 03 2013
%C A026792 This is the reverse-colexicographic order of integer partitions, or the reflected reverse-lexicographic order of reversed integer partitions. It is not reverse-lexicographic order (A080577), wherein we would have (3,1) before (2,2). - _Gus Wiseman_, May 12 2020
%H A026792 Robert Price, <a href="/A026792/b026792.txt">Table of n, a(n) for n = 1..3615, 15 rows.</a>
%H A026792 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A026792 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A026792 E.g. the partitions of 3 (3,2+1,1+1+1) appear as the string 3,2,1,1,1,1.
%e A026792 So the list begins:
%e A026792 1
%e A026792 2, 1, 1,
%e A026792 3, 2, 1, 1, 1, 1,
%e A026792 4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1,
%e A026792 5, 3, 2, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,
%e A026792 ...
%e A026792 From _Omar E. Pol_, Sep 03 2013: (Start)
%e A026792 Illustration of initial terms:
%e A026792 ---------------------------------
%e A026792 n  j     Diagram     Partition
%e A026792 ---------------------------------
%e A026792 .         _
%e A026792 1  1     |_|         1;
%e A026792 .         _ _
%e A026792 2  1     |_  |       2,
%e A026792 2  2     |_|_|       1, 1;
%e A026792 .         _ _ _
%e A026792 3  1     |_ _  |     3,
%e A026792 3  2     |_  | |     2, 1,
%e A026792 3  3     |_|_|_|     1, 1, 1;
%e A026792 .         _ _ _ _
%e A026792 4  1     |_ _    |   4,
%e A026792 4  2     |_ _|_  |   2, 2,
%e A026792 4  3     |_ _  | |   3, 1,
%e A026792 4  4     |_  | | |   2, 1, 1,
%e A026792 4  5     |_|_|_|_|   1, 1, 1, 1;
%e A026792 ...
%e A026792 (End)
%e A026792 From _Gus Wiseman_, May 12 2020: (Start)
%e A026792 This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows. Showing these partitions as their Heinz numbers gives A334436.
%e A026792                              0
%e A026792                             (1)
%e A026792                           (2)(11)
%e A026792                         (3)(21)(111)
%e A026792                    (4)(22)(31)(211)(1111)
%e A026792              (5)(32)(41)(221)(311)(2111)(11111)
%e A026792   (6)(33)(42)(222)(51)(321)(411)(2211)(3111)(21111)(111111)
%e A026792 (End)
%t A026792 revcolex[f_,c_]:=OrderedQ[PadRight[{Reverse[c],Reverse[f]}]];
%t A026792 Join@@Table[Sort[IntegerPartitions[n],revcolex],{n,0,8}] (* reverse-colexicographic order, _Gus Wiseman_, May 10 2020 *)
%t A026792 - or -
%t A026792 revlex[f_,c_]:=OrderedQ[PadRight[{c,f}]];
%t A026792 Reverse/@Join@@Table[Sort[Reverse/@IntegerPartitions[n],revlex],{n,0,8}] (* reflected reverse-lexicographic order, _Gus Wiseman_, May 12 2020 *)
%Y A026792 Cf. A026791, A228531.
%Y A026792 The reflected version for reversed partitions is A080577.
%Y A026792 The partition minima appear to be A182715.
%Y A026792 The graded reversed version is A211992.
%Y A026792 The version for compositions is A228351.
%Y A026792 The Heinz numbers of these partitions are A334436.
%Y A026792 Cf. A000041, A036036, A036037, A193073, A296150, A331581, A334301, A334435, A334437, A334439.
%K A026792 nonn,tabf
%O A026792 1,2
%A A026792 _Clark Kimberling_
%E A026792 Terms 81st, 83rd and 84th corrected by _Omar E. Pol_, Aug 16 2009