A334436 Heinz numbers of all reversed integer partitions sorted first by sum and then reverse-lexicographically.

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

Offset: 0

Author

Gus Wiseman, May 02 2020

Keywords

• nonn
• tabf

Comments

First differs from A334435 at a(22) = 27, A334435(22) = 22.

A permutation of the positive integers.

Reversed integer partitions are finite weakly increasing sequences of positive integers. For non-reversed partitions, see A129129 and A228531.

This is the so-called "Mathematica" order (A080577).

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.

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}               54: {1,2,2,2}
    3: {2}           25: {3,3}             44: {1,1,5}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           27: {2,2,2}           56: {1,1,1,4}
    6: {1,2}         22: {1,5}             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}             75: {2,3,3}
   14: {1,4}         33: {2,5}             63: {2,2,4}
   18: {1,2,2}       45: {2,2,3}           81: {2,2,2,2}
   20: {1,1,3}       26: {1,6}             34: {1,7}
   24: {1,1,1,2}     50: {1,3,3}           70: {1,3,4}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  27  22  30  28  36  40  48  64
  17  35  33  45  26  50  42  54  44  60  56  72  80  96 128
This corresponds to the following tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(12)(111)
        (4)(22)(13)(112)(1111)
  (5)(23)(14)(122)(113)(1112)(11111)
		

Links

• Wikiversity, Lexicographic and colexicographic order

Crossrefs

Row lengths are A000041.

Compositions under the same order are A066099 (triangle).

The version for non-reversed partitions is A129129.

The constructive version is A228531.

The lengths of these partitions are A333486.

The length-sensitive version is A334435.

The dual version (sum/lex) is A334437.

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 A215366.

Sorting partitions by Heinz number gives A296150.

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

Cf. A056239, A124734, A185974, A228100, A333219, A334301, A334302, A334433, A334434, A334438.

Programs

Mathematica

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

Formula

A001222(a(n)) = A333486(n).