|
|
A334439
|
|
Irregular triangle whose rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
|
|
43
|
|
|
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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 4, 3, 5, 1, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
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).
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.
|
|
LINKS
|
|
|
EXAMPLE
|
The sequence of all partitions begins:
() (32) (21111) (22111) (4211) (63)
(1) (311) (111111) (211111) (3311) (54)
(2) (221) (7) (1111111) (3221) (711)
(11) (2111) (61) (8) (2222) (621)
(3) (11111) (52) (71) (41111) (531)
(21) (6) (43) (62) (32111) (522)
(111) (51) (511) (53) (22211) (441)
(4) (42) (421) (44) (311111) (432)
(31) (33) (331) (611) (221111) (333)
(22) (411) (322) (521) (2111111) (6111)
(211) (321) (4111) (431) (11111111) (5211)
(1111) (222) (3211) (422) (9) (4311)
(5) (3111) (2221) (332) (81) (4221)
(41) (2211) (31111) (5111) (72) (3321)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.
0
(1)
(2)(11)
(3)(21)(111)
(4)(31)(22)(211)(1111)
(5)(41)(32)(311)(221)(2111)(11111)
Showing partitions as their Heinz numbers (see A334438) gives:
1
2
3 4
5 6 8
7 10 9 12 16
11 14 15 20 18 24 32
13 22 21 25 28 30 27 40 36 48 64
17 26 33 35 44 42 50 45 56 60 54 80 72 96 128
|
|
MATHEMATICA
|
revlensort[f_, c_]:=If[Length[f]!=Length[c], Length[f]<Length[c], OrderedQ[{c, f}]];
Join@@Table[Sort[IntegerPartitions[n], revlensort], {n, 0, 8}]
|
|
CROSSREFS
|
The version for colex instead of revlex is A036037.
Number of distinct elements in row n appears to be A103921(n).
The version for compositions is A296774.
The Abramowitz-Stegun version (sum/length/lex) is A334301.
The version for reversed partitions is A334302.
Taking Heinz numbers gives A334438.
The version with partitions reversed is A334442.
Lexicographically ordered reversed partitions are A026791.
Lexicographically ordered partitions are A193073.
Sorting partitions by Heinz number gives A296150.
Cf. A000041, A036036, A112798, A124734, A129129, A185974, A228100, A228531, A334433, A334435, A334436.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|