OFFSET
0,2
COMMENTS
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.
LINKS
EXAMPLE
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)
MATHEMATICA
lexsort[f_, c_]:=OrderedQ[PadRight[{f, c}]];
Table[Times@@Prime/@#&/@Reverse[Sort[Sort/@IntegerPartitions[n], lexsort]], {n, 0, 8}]
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.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 02 2020
STATUS
approved