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A334435
Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
32
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, 19, 49, 55, 39, 34, 75, 63, 70, 66, 52, 81, 90, 100, 84, 88, 108, 120, 112, 144, 160, 192, 256
OFFSET
0,2
COMMENTS
First differs from A334433 at a(75) = 99, A334433(75) = 98.
First differs from A334436 at a(22) = 22, A334436(22) = 27.
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.
FORMULA
A001222(a(n)) = A036043(n).
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}]
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.
Sequence in context: A215366 A333483 A334433 * A334436 A266195 A102530
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 02 2020
STATUS
approved