A334439 - OEIS

%I #15 Sep 22 2023 05:18:21

%S 1,2,1,1,3,2,1,1,1,1,4,3,1,2,2,2,1,1,1,1,1,1,5,4,1,3,2,3,1,1,2,2,1,2,

%T 1,1,1,1,1,1,1,1,6,5,1,4,2,3,3,4,1,1,3,2,1,2,2,2,3,1,1,1,2,2,1,1,2,1,

%U 1,1,1,1,1,1,1,1,1,7,6,1,5,2,4,3,5,1,1

%N Irregular triangle whose rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

%C First differs from A036037 for partitions of 9. Namely, this sequence has (5,2,2) before (4,4,1), while A036037 has (4,4,1) before (5,2,2).

%C This is the Abramowitz-Stegun ordering of integer partitions (A334301) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334302.

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

%e The sequence of all partitions begins:

%e   ()      (32)     (21111)   (22111)    (4211)      (63)

%e   (1)     (311)    (111111)  (211111)   (3311)      (54)

%e   (2)     (221)    (7)       (1111111)  (3221)      (711)

%e   (11)    (2111)   (61)      (8)        (2222)      (621)

%e   (3)     (11111)  (52)      (71)       (41111)     (531)

%e   (21)    (6)      (43)      (62)       (32111)     (522)

%e   (111)   (51)     (511)     (53)       (22211)     (441)

%e   (4)     (42)     (421)     (44)       (311111)    (432)

%e   (31)    (33)     (331)     (611)      (221111)    (333)

%e   (22)    (411)    (322)     (521)      (2111111)   (6111)

%e   (211)   (321)    (4111)    (431)      (11111111)  (5211)

%e   (1111)  (222)    (3211)    (422)      (9)         (4311)

%e   (5)     (3111)   (2221)    (332)      (81)        (4221)

%e   (41)    (2211)   (31111)   (5111)     (72)        (3321)

%e This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.

%e                   0

%e                  (1)

%e                (2)(11)

%e              (3)(21)(111)

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

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

%e Showing partitions as their Heinz numbers (see A334438) gives:

%e    1

%e    2

%e    3   4

%e    5   6   8

%e    7  10   9  12  16

%e   11  14  15  20  18  24  32

%e   13  22  21  25  28  30  27  40  36  48  64

%e   17  26  33  35  44  42  50  45  56  60  54  80  72  96 128

%t revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];

%t Join@@Table[Sort[IntegerPartitions[n],revlensort],{n,0,8}]

%Y The version for colex instead of revlex is A036037.

%Y Row lengths are A036043.

%Y Ignoring length gives A080577.

%Y Number of distinct elements in row n appears to be A103921(n).

%Y The version for compositions is A296774.

%Y The Abramowitz-Stegun version (sum/length/lex) is A334301.

%Y The version for reversed partitions is A334302.

%Y Taking Heinz numbers gives A334438.

%Y The version with partitions reversed is A334442.

%Y Lexicographically ordered reversed partitions are A026791.

%Y Lexicographically ordered partitions are A193073.

%Y Sorting partitions by Heinz number gives A296150.

%Y Cf. A000041, A036036, A112798, A124734, A129129, A185974, A228100, A228531, A334433, A334435, A334436.

%K nonn,tabf

%O 0,2

%A _Gus Wiseman_, May 03 2020