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A334442
Irregular triangle whose reversed rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
19
1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 5, 1, 4, 2, 3, 1, 1, 3, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 6, 1, 5, 2, 4, 3, 3, 1, 1, 4, 1, 2, 3, 2, 2, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7, 1, 6, 2, 5, 3, 4, 1, 1, 5
OFFSET
0,2
COMMENTS
First differs from A036036 for reversed partitions of 9. Namely, this sequence has (2,2,5) before (1,4,4), while A036036 has (1,4,4) before (2,2,5).
EXAMPLE
The sequence of all partitions begins:
() (2,3) (1,1,1,1,2) (1,1,1,2,2)
(1) (1,1,3) (1,1,1,1,1,1) (1,1,1,1,1,2)
(2) (1,2,2) (7) (1,1,1,1,1,1,1)
(1,1) (1,1,1,2) (1,6) (8)
(3) (1,1,1,1,1) (2,5) (1,7)
(1,2) (6) (3,4) (2,6)
(1,1,1) (1,5) (1,1,5) (3,5)
(4) (2,4) (1,2,4) (4,4)
(1,3) (3,3) (1,3,3) (1,1,6)
(2,2) (1,1,4) (2,2,3) (1,2,5)
(1,1,2) (1,2,3) (1,1,1,4) (1,3,4)
(1,1,1,1) (2,2,2) (1,1,2,3) (2,2,4)
(5) (1,1,1,3) (1,2,2,2) (2,3,3)
(1,4) (1,1,2,2) (1,1,1,1,3) (1,1,1,5)
This sequence can also be interpreted as the following triangle:
0
(1)
(2)(11)
(3)(12)(111)
(4)(13)(22)(112)(1111)
(5)(14)(23)(113)(122)(1112)(11111)
Taking Heinz numbers (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@@Reverse/@Join@@Table[Sort[IntegerPartitions[n], revlensort], {n, 0, 8}]
PROG
(PARI) A334442_row(n)=vecsort(partitions(n), p->concat(#p, -Vecrev(p))) \\ Rows of triangle defined in EXAMPLE (all partitions of n). Wrap into [Vec(p)|p<-...] to avoid "Vecsmall". - M. F. Hasler, May 14 2020
CROSSREFS
Row lengths are A036043.
The version for reversed partitions is A334301.
The version for colex instead of revlex is A334302.
Taking Heinz numbers gives A334438.
The version with rows reversed is A334439.
Ignoring length gives A335122.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex order (sum/length/colex) are A036037.
Reverse-lexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Sorting partitions by Heinz number gives A296150.
Sequence in context: A094293 A338156 A335122 * A036036 A344091 A334302
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 07 2020
STATUS
approved