A334434 Heinz number of the n-th integer partition in graded lexicographic order.

1, 2, 4, 3, 8, 6, 5, 16, 12, 9, 10, 7, 32, 24, 18, 20, 15, 14, 11, 64, 48, 36, 27, 40, 30, 25, 28, 21, 22, 13, 128, 96, 72, 54, 80, 60, 45, 50, 56, 42, 35, 44, 33, 26, 17, 256, 192, 144, 108, 81, 160, 120, 90, 100, 75, 112, 84, 63, 70, 49, 88, 66, 55, 52, 39, 34, 19

Offset

0,2

Comments

A permutation of the positive integers.

This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).

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},{4,3},{8,6,5},...}, so offset is 0.

Links

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

OEIS Wiki, Orderings of partitions

Wikiversity, Lexicographic and colexicographic order

Formula

A001222(a(n)) appears to be A049085(n).

Example

The sequence of terms together with their prime indices begins:
    1: {}              11: {5}                 45: {2,2,3}
    2: {1}             64: {1,1,1,1,1,1}       50: {1,3,3}
    4: {1,1}           48: {1,1,1,1,2}         56: {1,1,1,4}
    3: {2}             36: {1,1,2,2}           42: {1,2,4}
    8: {1,1,1}         27: {2,2,2}             35: {3,4}
    6: {1,2}           40: {1,1,1,3}           44: {1,1,5}
    5: {3}             30: {1,2,3}             33: {2,5}
   16: {1,1,1,1}       25: {3,3}               26: {1,6}
   12: {1,1,2}         28: {1,1,4}             17: {7}
    9: {2,2}           21: {2,4}              256: {1,1,1,1,1,1,1,1}
   10: {1,3}           22: {1,5}              192: {1,1,1,1,1,1,2}
    7: {4}             13: {6}                144: {1,1,1,1,2,2}
   32: {1,1,1,1,1}    128: {1,1,1,1,1,1,1}    108: {1,1,2,2,2}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}       81: {2,2,2,2}
   18: {1,2,2}         72: {1,1,1,2,2}        160: {1,1,1,1,1,3}
   20: {1,1,3}         54: {1,2,2,2}          120: {1,1,1,2,3}
   15: {2,3}           80: {1,1,1,1,3}         90: {1,2,2,3}
   14: {1,4}           60: {1,1,2,3}          100: {1,1,3,3}
Triangle begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  15  14  11
   64  48  36  27  40  30  25  28  21  22  13
  128  96  72  54  80  60  45  50  56  42  35  44  33  26  17
This corresponds to the tetrangle:
                  0
                 (1)
               (11)(2)
             (111)(21)(3)
        (1111)(211)(22)(31)(4)
  (11111)(2111)(221)(311)(32)(41)(5)

Mathematica

lexsort[f_, c_]:=OrderedQ[PadRight[{f, c}]];
Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n], lexsort], {n, 0, 8}]
- or -
Join@@Table[Times@@Prime/@#&/@Reverse[IntegerPartitions[n]], {n, 0, 8}]

Crossrefs

Row lengths are A000041.

The dual version (sum/revlex) is A129129.

The constructive version is A193073.

Compositions under the same order are A228351.

The length-sensitive version is A334433.

The version for reversed (weakly increasing) partitions is A334437.

Lexicographically ordered reversed partitions are A026791.

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

Reverse-lexicographically ordered partitions are A080577.

Sorting reversed partitions by Heinz number gives A112798.

Graded Heinz numbers are A215366.

Sorting partitions by Heinz number gives A296150.

Cf. A036037, A049085, A056239, A066099, A185974, A211992, A228100, A228531, A333219, A334301, A334302, A334435, A334436, A334438, A334439.

Sequence in context: A260431 A376738 A333484 * A333485 A246166 A334437

Adjacent sequences: A334431 A334432 A334433 * A334435 A334436 A334437

Keyword

nonn,tabf

Author

Gus Wiseman, May 01 2020

Status

approved