A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

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

Offset

0,2

Links

Table of n, a(n) for n=0..86.

OEIS Wiki, Orderings of partitions

Wikiversity, Lexicographic and colexicographic order

Example

The sequence of all reversed partitions begins:
  ()         (1,4)        (1,1,1,1,2)
  (1)        (1,2,2)      (1,1,1,1,1,1)
  (2)        (1,1,3)      (7)
  (1,1)      (1,1,1,2)    (3,4)
  (3)        (1,1,1,1,1)  (2,5)
  (1,2)      (6)          (1,6)
  (1,1,1)    (3,3)        (2,2,3)
  (4)        (2,4)        (1,3,3)
  (2,2)      (1,5)        (1,2,4)
  (1,3)      (2,2,2)      (1,1,5)
  (1,1,2)    (1,2,3)      (1,2,2,2)
  (1,1,1,1)  (1,1,4)      (1,1,2,3)
  (5)        (1,1,2,2)    (1,1,1,4)
  (2,3)      (1,1,1,3)    (1,1,1,2,2)
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) (1,1)
                    (3) (1,2) (1,1,1)
            (4) (2,2) (1,3) (1,1,2) (1,1,1,1)
  (5) (2,3) (1,4) (1,2,2) (1,1,3) (1,1,1,2) (1,1,1,1,1)
Showing partitions as their Heinz numbers (see A334435) gives:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128

Mathematica

revlensort[f_, c_]:=If[Length[f]!=Length[c], Length[f]<Length[c], OrderedQ[{c, f}]];
Join@@Table[Sort[Sort/@IntegerPartitions[n], revlensort], {n, 0, 8}]

Crossrefs

Row lengths are A036043.

Lexicographically ordered reversed partitions are A026791.

The dual ordering (sum/length/lex) of reversed partitions is A036036.

Reverse-lexicographically ordered partitions are A080577.

Sorting reversed partitions by Heinz number gives A112798.

Lexicographically ordered partitions are A193073.

Graded Heinz numbers are A215366.

Ignoring length gives A228531.

Sorting partitions by Heinz number gives A296150.

The version for compositions is A296774.

The dual ordering (sum/length/lex) of non-reversed partitions is A334301.

Taking Heinz numbers gives A334435.

The version for regular (non-reversed) partitions is A334439 (not A036037).

Cf. A000041, A048793, A066099, A080576, A124734, A162247, A211992, A228100, A228351.

Sequence in context: A334442 A036036 A344091 * A228531 A360056 A244316

Adjacent sequences: A334299 A334300 A334301 * A334303 A334304 A334305

Keyword

nonn,tabf

Author

Gus Wiseman, Apr 30 2020

Status

approved