login
A334438
Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
28
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, 19, 34, 39, 55, 49, 52, 66, 70, 63, 75, 88, 84, 100, 90, 81, 112, 120, 108, 160, 144, 192, 256
OFFSET
0,2
COMMENTS
First differs from A185974 shifted left once at a(76) = 99, A185974(75) = 98.
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.
FORMULA
A001221(a(n)) = A103921(n).
A001222(a(n)) = A036043(n).
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}]
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.
Sequence in context: A333658 A337598 A333221 * A185974 A129129 A114622
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 03 2020
STATUS
approved