OFFSET
0,2
COMMENTS
A permutation of the positive integers.
This is the Abramowitz-Stegun ordering of integer partitions (A334433) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334435.
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.
This sequence has partitions in the order (5,3,1), (5,2,2), (4,4,1), while A185974 has them in the order (5,3,1), (4,4,1), (5,2,2). - Andrew Howroyd, Oct 06 2025
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} 50: {1,3,3}
2: {1} 13: {6} 45: {2,2,3}
3: {2} 22: {1,5} 56: {1,1,1,4}
4: {1,1} 21: {2,4} 60: {1,1,2,3}
5: {3} 25: {3,3} 54: {1,2,2,2}
6: {1,2} 28: {1,1,4} 80: {1,1,1,1,3}
8: {1,1,1} 30: {1,2,3} 72: {1,1,1,2,2}
7: {4} 27: {2,2,2} 96: {1,1,1,1,1,2}
10: {1,3} 40: {1,1,1,3} 128: {1,1,1,1,1,1,1}
9: {2,2} 36: {1,1,2,2} 19: {8}
12: {1,1,2} 48: {1,1,1,1,2} 34: {1,7}
16: {1,1,1,1} 64: {1,1,1,1,1,1} 39: {2,6}
11: {5} 17: {7} 55: {3,5}
14: {1,4} 26: {1,6} 49: {4,4}
15: {2,3} 33: {2,5} 52: {1,1,6}
20: {1,1,3} 35: {3,4} 66: {1,2,5}
18: {1,2,2} 44: {1,1,5} 70: {1,3,4}
24: {1,1,1,2} 42: {1,2,4} 63: {2,2,4}
Triangle begins:
1
2
3 4
5 6 8
7 10 9 12 16
11 14 15 20 18 24 32
13 22 21 25 28 30 27 40 36 48 64
17 26 33 35 44 42 50 45 56 60 54 80 72 96 128
This corresponds to the following tetrangle:
0
(1)
(2)(11)
(3)(21)(111)
(4)(31)(22)(211)(1111)
(5)(41)(32)(311)(221)(2111)(11111)
MATHEMATICA
revlensort[f_, c_]:=If[Length[f]!=Length[c], Length[f]<Length[c], OrderedQ[{c, f}]];
Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n], revlensort], {n, 0, 8}]
PROG
(PARI)
C(sig)=prod(k=1, #sig, prime(sig[k]))
cmpLenLexDesc(x, y)=my(d=cmp(#x, #y)); if(d, d, -lex(x, y))
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)], cmpLenLexDesc))}
{ for(n=0, 9, print(Row(n))) } \\ Andrew Howroyd, Oct 06 2025
CROSSREFS
Row lengths are A000041.
Ignoring length gives A129129.
Compositions under the same order are A296774 (triangle).
The dual version (sum/length/lex) is A334433.
The version for reversed partitions is A334435.
The constructive version is A334439 (triangle).
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 given by A215366.
Sorting partitions by Heinz number gives A296150.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 03 2020
EXTENSIONS
a(67) onwards from Andrew Howroyd, Oct 06 2025
STATUS
approved
