A334439 Irregular triangle whose rows are all integer partitions of n sorted first by sum, then by length, and finally reverse-lexicographically.

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

Offset

1,2

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

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

Wikiversity, Lexicographic and colexicographic order

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

Maple

A334439row := proc(n) local b;  # after Alois P. Heinz
b := (n, i)-> ifelse(n = 0 or i = 1, [[1$n]],
[map(x -> [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
sort([map(x-> x[], [b(n$2)])[]], length) end:
seq(print(A334439row(n)), n = 0..10):  # Peter Luschny, Apr 29 2026

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}]

Prog

(PARI)
cmpLenLexDesc(x, y)=my(d=cmp(#x, #y)); if(d, d, -lex(x, y))
Row(n)={concat(vecsort([Vecrev(p) | p<-partitions(n)], cmpLenLexDesc))}
{ for(n=1, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 06 2025
(Python)
def A334439row(n):
    def b(n, i):
        if n == 0: yield []
        elif i == 1: yield [1] * n
        else:
            if n >= i: # Prepend 'i' to partitions of n-i with max part min(n-i, i)
                for p in b(n - i, min(n - i, i)): yield [i] + p
            yield from b(n, i - 1) # Yield partitions of n with max part i-1
    if n == 0: return [[]]
    partitions = list(b(n, n))
    partitions.sort(key=len)
    return partitions  # Peter Luschny, May 01 2026

Crossrefs

The version for colex instead of revlex is A036037.

Row lengths are A036043.

Ignoring length gives A080577.

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.

Sequence in context: A139100 A237982 A239512 * A036037 A181317 A330370

Adjacent sequences: A334436 A334437 A334438 * A334440 A334441 A334442

Keyword

nonn,tabf

Author

Gus Wiseman, May 03 2020

Extensions

Offset corrected by Andrew Howroyd, Oct 06 2025

Status

approved