OFFSET
0,2
COMMENTS
A permutation of the positive integers.
Reversed integer partitions are finite weakly increasing sequences of positive integers.
This is the Abramowitz-Stegun ordering of reversed partitions (A185974) except that the finer order is reverse-lexicographic instead of lexicographic. The version for non-reversed partitions is A334438.
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
Wikiversity, Lexicographic and colexicographic order
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} 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 following tetrangle:
0
(1)
(2)(11)
(3)(12)(111)
(4)(22)(13)(112)(1111)
(5)(23)(14)(122)(113)(1112)(11111)
MATHEMATICA
revlensort[f_, c_]:=If[Length[f]!=Length[c], Length[f]<Length[c], OrderedQ[{c, f}]];
Table[Times@@Prime/@#&/@Sort[Sort/@IntegerPartitions[n], revlensort], {n, 0, 8}]
PROG
(PARI)
cmpLenLexDesc(x, y)=my(d=cmp(#x, #y)); if(d, d, -lex(x, y))
C(sig)=prod(k=1, #sig, prime(sig[k]))
Row(n)={apply(C, vecsort([p | p<-partitions(n)], cmpLenLexDesc))}
{ for(n=0, 9, print(Row(n))) } \\ Andrew Howroyd, Oct 07 2025
CROSSREFS
Row lengths are A000041.
The dual version (sum/length/lex) is A185974.
Compositions under the same order are A296774 (triangle).
The constructive version is A334302.
Ignoring length gives A334436.
The version for non-reversed partitions is A334438.
Partitions in this order (sum/length/revlex) are A334439.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex 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 (sum/colex) order are A211992.
Graded Heinz numbers are given by A215366.
Sorting partitions by Heinz number gives A296150.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 02 2020
EXTENSIONS
a(67) onwards from Andrew Howroyd, Oct 07 2025
STATUS
approved
