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%I #20 Sep 22 2023 07:53:08
%S 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,
%T 36,40,48,64,17,35,33,26,45,50,42,44,54,60,56,72,80,96,128,19,49,55,
%U 39,34,75,63,70,66,52,81,90,100,84,88,108,120,112,144,160,192,256
%N Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
%C First differs from A334433 at a(75) = 99, A334433(75) = 98.
%C First differs from A334436 at a(22) = 22, A334436(22) = 27.
%C A permutation of the positive integers.
%C Reversed integer partitions are finite weakly increasing sequences of positive integers.
%C 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.
%C 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.
%C As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.
%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A001222(a(n)) = A036043(n).
%e The sequence of terms together with their prime indices begins:
%e 1: {} 32: {1,1,1,1,1} 42: {1,2,4}
%e 2: {1} 13: {6} 44: {1,1,5}
%e 3: {2} 25: {3,3} 54: {1,2,2,2}
%e 4: {1,1} 21: {2,4} 60: {1,1,2,3}
%e 5: {3} 22: {1,5} 56: {1,1,1,4}
%e 6: {1,2} 27: {2,2,2} 72: {1,1,1,2,2}
%e 8: {1,1,1} 30: {1,2,3} 80: {1,1,1,1,3}
%e 7: {4} 28: {1,1,4} 96: {1,1,1,1,1,2}
%e 9: {2,2} 36: {1,1,2,2} 128: {1,1,1,1,1,1,1}
%e 10: {1,3} 40: {1,1,1,3} 19: {8}
%e 12: {1,1,2} 48: {1,1,1,1,2} 49: {4,4}
%e 16: {1,1,1,1} 64: {1,1,1,1,1,1} 55: {3,5}
%e 11: {5} 17: {7} 39: {2,6}
%e 15: {2,3} 35: {3,4} 34: {1,7}
%e 14: {1,4} 33: {2,5} 75: {2,3,3}
%e 18: {1,2,2} 26: {1,6} 63: {2,2,4}
%e 20: {1,1,3} 45: {2,2,3} 70: {1,3,4}
%e 24: {1,1,1,2} 50: {1,3,3} 66: {1,2,5}
%e Triangle begins:
%e 1
%e 2
%e 3 4
%e 5 6 8
%e 7 9 10 12 16
%e 11 15 14 18 20 24 32
%e 13 25 21 22 27 30 28 36 40 48 64
%e 17 35 33 26 45 50 42 44 54 60 56 72 80 96 128
%e This corresponds to the following tetrangle:
%e 0
%e (1)
%e (2)(11)
%e (3)(12)(111)
%e (4)(22)(13)(112)(1111)
%e (5)(23)(14)(122)(113)(1112)(11111)
%t revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];
%t Table[Times@@Prime/@#&/@Sort[Sort/@IntegerPartitions[n],revlensort],{n,0,8}]
%Y Row lengths are A000041.
%Y The dual version (sum/length/lex) is A185974.
%Y Compositions under the same order are A296774 (triangle).
%Y The constructive version is A334302.
%Y Ignoring length gives A334436.
%Y The version for non-reversed partitions is A334438.
%Y Partitions in this order (sum/length/revlex) are A334439.
%Y Lexicographically ordered reversed partitions are A026791.
%Y Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y Partitions in increasing-length colex order (sum/length/colex) are A036037.
%Y Reverse-lexicographically ordered partitions are A080577.
%Y Sorting reversed partitions by Heinz number gives A112798.
%Y Graded lexicographically ordered partitions are A193073.
%Y Partitions in colexicographic (sum/colex) order are A211992.
%Y Graded Heinz numbers are given by A215366.
%Y Sorting partitions by Heinz number gives A296150.
%Y Cf. A056239, A124734, A129129, A228100, A228531, A333219, A333220, A334301, A334433, A334434, A334437.
%K nonn,tabf
%O 0,2
%A _Gus Wiseman_, May 02 2020
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