A334437 Heinz number of the n-th reversed integer partition in graded lexicographical order.

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

Offset: 0

Author

Gus Wiseman, May 03 2020

Keywords

• nonn
• tabf

Comments

A permutation of the positive integers.

Reversed integer partitions are finite weakly increasing sequences of positive integers. The non-reversed version is A334434.

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

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

Also Heinz numbers of partitions in colexicographic order (cf. A211992).

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

Examples

			The sequence of terms together with their prime indices begins:
    1: {}              11: {5}                 44: {1,1,5}
    2: {1}             64: {1,1,1,1,1,1}       54: {1,2,2,2}
    4: {1,1}           48: {1,1,1,1,2}         42: {1,2,4}
    3: {2}             40: {1,1,1,3}           50: {1,3,3}
    8: {1,1,1}         36: {1,1,2,2}           26: {1,6}
    6: {1,2}           28: {1,1,4}             45: {2,2,3}
    5: {3}             30: {1,2,3}             33: {2,5}
   16: {1,1,1,1}       22: {1,5}               35: {3,4}
   12: {1,1,2}         27: {2,2,2}             17: {7}
   10: {1,3}           21: {2,4}              256: {1,1,1,1,1,1,1,1}
    9: {2,2}           25: {3,3}              192: {1,1,1,1,1,1,2}
    7: {4}             13: {6}                160: {1,1,1,1,1,3}
   32: {1,1,1,1,1}    128: {1,1,1,1,1,1,1}    144: {1,1,1,1,2,2}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      112: {1,1,1,1,4}
   20: {1,1,3}         80: {1,1,1,1,3}        120: {1,1,1,2,3}
   18: {1,2,2}         72: {1,1,1,2,2}         88: {1,1,1,5}
   14: {1,4}           56: {1,1,1,4}          108: {1,1,2,2,2}
   15: {2,3}           60: {1,1,2,3}           84: {1,1,2,4}
Triangle begins:
    1
    2
    4   3
    8   6   5
   16  12  10   9   7
   32  24  20  18  14  15  11
   64  48  40  36  28  30  22  27  21  25  13
  128  96  80  72  56  60  44  54  42  50  26  45  33  35  17
This corresponds to the following tetrangle:
                  0
                 (1)
               (11)(2)
             (111)(12)(3)
        (1111)(112)(13)(22)(4)
  (11111)(1112)(113)(122)(14)(23)(5)
		

Links

• Wikiversity, Lexicographic and colexicographic order

Crossrefs

Row lengths are A000041.

The constructive version is A026791 (triangle).

The length-sensitive version is A185974.

Compositions under the same order are A228351 (triangle).

The version for non-reversed partitions is A334434.

The dual version (sum/revlex) is A334436.

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.

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

Graded Heinz numbers are given by A215366.

Sorting partitions by Heinz number gives A296150.

Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.

Cf. A056239, A066099, A129129, A228531, A333219, A333220, A334301, A334302, A334433, A334435, A334438.

Programs

Mathematica

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

Formula

A001222(a(n)) = A193173(n).