A335122 Irregular triangle whose reversed rows are all integer partitions in graded reverse-lexicographic order.

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

Offset

1,2

Comments

First differs from A036036 for partitions of 6.

First differs from A334442 for partitions of 6.

Also reversed partitions in reverse-colexicographic order.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..3615 (rows 1..15)

OEIS Wiki, Orderings of partitions

Wikiversity, Lexicographic and colexicographic order

Example

The sequence of all reversed partitions begins:
  ()         (1,1,3)        (7)              (8)
  (1)        (1,2,2)        (1,6)            (1,7)
  (2)        (1,1,1,2)      (2,5)            (2,6)
  (1,1)      (1,1,1,1,1)    (1,1,5)          (1,1,6)
  (3)        (6)            (3,4)            (3,5)
  (1,2)      (1,5)          (1,2,4)          (1,2,5)
  (1,1,1)    (2,4)          (1,1,1,4)        (1,1,1,5)
  (4)        (1,1,4)        (1,3,3)          (4,4)
  (1,3)      (3,3)          (2,2,3)          (1,3,4)
  (2,2)      (1,2,3)        (1,1,2,3)        (2,2,4)
  (1,1,2)    (1,1,1,3)      (1,1,1,1,3)      (1,1,2,4)
  (1,1,1,1)  (2,2,2)        (1,2,2,2)        (1,1,1,1,4)
  (5)        (1,1,2,2)      (1,1,1,2,2)      (2,3,3)
  (1,4)      (1,1,1,1,2)    (1,1,1,1,1,2)    (1,1,3,3)
  (2,3)      (1,1,1,1,1,1)  (1,1,1,1,1,1,1)  (1,2,2,3)
We have the following tetrangle of reversed partitions:
                             0
                            (1)
                          (2)(11)
                        (3)(12)(111)
                   (4)(13)(22)(112)(1111)
             (5)(14)(23)(113)(122)(1112)(11111)
  (6)(15)(24)(114)(33)(123)(1113)(222)(1122)(11112)(111111)

Mathematica

revlexsort[f_, c_]:=OrderedQ[PadRight[{c, f}]];
Reverse/@Join@@Table[Sort[IntegerPartitions[n], revlexsort], {n, 0, 8}]

Prog

(PARI) Row(n)={concat(apply(vecsort, vecsort([Vecrev(p) | p<-partitions(n)], , 4)))}
{ for(n=1, 6, print(row(n))) } \\ Andrew Howroyd, Oct 07 2025

Crossrefs

Row lengths are A000041.

The version for reversed partitions is A026792.

The version for colex instead of revlex is A026791.

The version for lex instead of revlex is A080576.

The non-reflected version is A080577.

The number of distinct parts is A115623.

Taking Heinz numbers gives A129129.

The version for compositions is A228351.

Partition lengths are A238966.

Partition maxima are A331581.

The length-sensitive version is A334442.

Lexicographically ordered partitions are A193073.

Partitions in colexicographic order are A211992.

Cf. A036036, A036037, A112798, A129129, A228531, A296774, A334301, A334302, A334435, A334436, A334438, A334439.

Sequence in context: A057898 A094293 A338156 * A334442 A036036 A344091

Adjacent sequences: A335119 A335120 A335121 * A335123 A335124 A335125

Keyword

nonn,tabf

Author

Gus Wiseman, May 24 2020

Extensions

Offset corrected by Andrew Howroyd, Oct 07 2025

Status

approved