A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

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

Offset: 0

Author

Gus Wiseman, Apr 30 2020

Keywords

• nonn
• tabf

Comments

First differs from A334435 at a(75) = 99, A334435(75) = 98.

A permutation of the positive integers.

This is the Abramowitz-Stegun ordering of integer partitions when the parts are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A185974.

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.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       42: {1,2,4}
    2: {1}           13: {6}               44: {1,1,5}
    3: {2}           25: {3,3}             54: {1,2,2,2}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           22: {1,5}             56: {1,1,1,4}
    6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
    8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
    7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
    9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
   10: {1,3}         40: {1,1,1,3}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
   11: {5}           17: {7}               39: {2,6}
   15: {2,3}         35: {3,4}             34: {1,7}
   14: {1,4}         33: {2,5}             75: {2,3,3}
   18: {1,2,2}       26: {1,6}             63: {2,2,4}
   20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
   24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
Triangle begins:
   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
This corresponds to the tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(21)(111)
        (4)(22)(31)(211)(1111)
  (5)(32)(41)(221)(311)(2111)(11111)
		

Links

• OEIS Wiki, Orderings of partitions
• Wikiversity, Lexicographic and colexicographic order

Crossrefs

Row lengths are A000041.

Compositions under the same order are A124734 (triangle).

The version for reversed (weakly increasing) partitions is A185974.

The constructive version is A334301.

Ignoring length gives A334434, or A334437 for reversed partitions.

The dual version (sum/length/revlex) is A334438.

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.

Graded reverse-lexicographically ordered partitions are A080577.

Sorting reversed partitions by Heinz number gives A112798.

Graded lexicographically ordered partitions are A193073.

Graded Heinz numbers are A215366.

Sorting partitions by Heinz number gives A296150.

Partitions in increasing-length reverse-lexicographic order (sum/length/revlex) are A334439 (not A036037).

Cf. A026791, A036043, A048793, A056239, A129129, A211992, A228351, A228531, A334302, A334435, A334436.

Programs

Mathematica

Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n]],{n,0,8}]

Formula

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