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List of juxtaposed reverse-lexicographically ordered partitions of the positive integers.
45

%I #58 Aug 06 2024 00:02:51

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

%C 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 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 The equivalent sequence for compositions (ordered partitions) is A228351. - _Omar E. Pol_, Sep 03 2013

%C 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 Robert Price, <a href="/A026792/b026792.txt">Table of n, a(n) for n = 1..3615, 15 rows.</a>

%H OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>

%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>

%e E.g. the partitions of 3 (3,2+1,1+1+1) appear as the string 3,2,1,1,1,1.

%e So the list begins:

%e 1

%e 2, 1, 1,

%e 3, 2, 1, 1, 1, 1,

%e 4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1,

%e 5, 3, 2, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,

%e ...

%e From _Omar E. Pol_, Sep 03 2013: (Start)

%e Illustration of initial terms:

%e ---------------------------------

%e n j Diagram Partition

%e ---------------------------------

%e . _

%e 1 1 |_| 1;

%e . _ _

%e 2 1 |_ | 2,

%e 2 2 |_|_| 1, 1;

%e . _ _ _

%e 3 1 |_ _ | 3,

%e 3 2 |_ | | 2, 1,

%e 3 3 |_|_|_| 1, 1, 1;

%e . _ _ _ _

%e 4 1 |_ _ | 4,

%e 4 2 |_ _|_ | 2, 2,

%e 4 3 |_ _ | | 3, 1,

%e 4 4 |_ | | | 2, 1, 1,

%e 4 5 |_|_|_|_| 1, 1, 1, 1;

%e ...

%e (End)

%e From _Gus Wiseman_, May 12 2020: (Start)

%e 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 0

%e (1)

%e (2)(11)

%e (3)(21)(111)

%e (4)(22)(31)(211)(1111)

%e (5)(32)(41)(221)(311)(2111)(11111)

%e (6)(33)(42)(222)(51)(321)(411)(2211)(3111)(21111)(111111)

%e (End)

%t revcolex[f_,c_]:=OrderedQ[PadRight[{Reverse[c],Reverse[f]}]];

%t Join@@Table[Sort[IntegerPartitions[n],revcolex],{n,0,8}] (* reverse-colexicographic order, _Gus Wiseman_, May 10 2020 *)

%t - or -

%t revlex[f_,c_]:=OrderedQ[PadRight[{c,f}]];

%t Reverse/@Join@@Table[Sort[Reverse/@IntegerPartitions[n],revlex],{n,0,8}] (* reflected reverse-lexicographic order, _Gus Wiseman_, May 12 2020 *)

%Y Cf. A026791, A228531.

%Y The reflected version for reversed partitions is A080577.

%Y The partition minima appear to be A182715.

%Y The graded reversed version is A211992.

%Y The version for compositions is A228351.

%Y The Heinz numbers of these partitions are A334436.

%Y Cf. A000041, A036036, A036037, A193073, A296150, A331581, A334301, A334435, A334437, A334439.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_

%E Terms 81st, 83rd and 84th corrected by _Omar E. Pol_, Aug 16 2009