A334438 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

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, 19, 34, 39, 55, 49, 52, 66, 70, 63, 75, 88, 84, 100, 90, 81, 112, 120, 108, 160, 144, 192, 256

Offset

0,2

Comments

First differs from A185974 shifted left once at a(76) = 99, A185974(75) = 98.

A permutation of the positive integers.

This is the Abramowitz-Stegun ordering of integer partitions (A334433) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334435.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

Links

Table of n, a(n) for n=0..66.

Wikiversity, Lexicographic and colexicographic order

Formula

A001221(a(n)) = A103921(n).

A001222(a(n)) = A036043(n).

Example

The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       50: {1,3,3}
    2: {1}           13: {6}               45: {2,2,3}
    3: {2}           22: {1,5}             56: {1,1,1,4}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           25: {3,3}             54: {1,2,2,2}
    6: {1,2}         28: {1,1,4}           80: {1,1,1,1,3}
    8: {1,1,1}       30: {1,2,3}           72: {1,1,1,2,2}
    7: {4}           27: {2,2,2}           96: {1,1,1,1,1,2}
   10: {1,3}         40: {1,1,1,3}        128: {1,1,1,1,1,1,1}
    9: {2,2}         36: {1,1,2,2}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       34: {1,7}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     39: {2,6}
   11: {5}           17: {7}               55: {3,5}
   14: {1,4}         26: {1,6}             49: {4,4}
   15: {2,3}         33: {2,5}             52: {1,1,6}
   20: {1,1,3}       35: {3,4}             66: {1,2,5}
   18: {1,2,2}       44: {1,1,5}           70: {1,3,4}
   24: {1,1,1,2}     42: {1,2,4}           63: {2,2,4}
Triangle begins:
   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
This corresponds to the following tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(21)(111)
        (4)(31)(22)(211)(1111)
  (5)(41)(32)(311)(221)(2111)(11111)

Mathematica

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

Crossrefs

Row lengths are A000041.

Ignoring length gives A129129.

Compositions under the same order are A296774 (triangle).

The dual version (sum/length/lex) is A334433.

The version for reversed partitions is A334435.

The constructive version is A334439 (triangle).

Lexicographically ordered reversed partitions are A026791.

Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.

Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.

Reverse-lexicographically ordered partitions are A080577.

Sorting reversed partitions by Heinz number gives A112798.

Graded lexicographically ordered partitions are A193073.

Partitions in colexicographic order (sum/colex) are A211992.

Graded Heinz numbers are given by A215366.

Sorting partitions by Heinz number gives A296150.

Cf. A056239, A066099, A124734, A185974, A228100, A228531, A333219, A334301, A334302, A334434, A334436, A334437.

Sequence in context: A333658 A337598 A333221 * A185974 A129129 A114622

Adjacent sequences: A334435 A334436 A334437 * A334439 A334440 A334441

Keyword

nonn,tabf

Author

Gus Wiseman, May 03 2020

Status

approved