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%I #17 Sep 22 2023 07:58:47
%S 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,
%T 40,36,48,64,17,26,33,35,44,42,50,45,56,60,54,80,72,96,128,19,34,39,
%U 55,49,52,66,70,63,75,88,84,100,90,81,112,120,108,160,144,192,256
%N Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
%C First differs from A185974 shifted left once at a(76) = 99, A185974(75) = 98.
%C A permutation of the positive integers.
%C 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.
%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 A001221(a(n)) = A103921(n).
%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} 50: {1,3,3}
%e 2: {1} 13: {6} 45: {2,2,3}
%e 3: {2} 22: {1,5} 56: {1,1,1,4}
%e 4: {1,1} 21: {2,4} 60: {1,1,2,3}
%e 5: {3} 25: {3,3} 54: {1,2,2,2}
%e 6: {1,2} 28: {1,1,4} 80: {1,1,1,1,3}
%e 8: {1,1,1} 30: {1,2,3} 72: {1,1,1,2,2}
%e 7: {4} 27: {2,2,2} 96: {1,1,1,1,1,2}
%e 10: {1,3} 40: {1,1,1,3} 128: {1,1,1,1,1,1,1}
%e 9: {2,2} 36: {1,1,2,2} 19: {8}
%e 12: {1,1,2} 48: {1,1,1,1,2} 34: {1,7}
%e 16: {1,1,1,1} 64: {1,1,1,1,1,1} 39: {2,6}
%e 11: {5} 17: {7} 55: {3,5}
%e 14: {1,4} 26: {1,6} 49: {4,4}
%e 15: {2,3} 33: {2,5} 52: {1,1,6}
%e 20: {1,1,3} 35: {3,4} 66: {1,2,5}
%e 18: {1,2,2} 44: {1,1,5} 70: {1,3,4}
%e 24: {1,1,1,2} 42: {1,2,4} 63: {2,2,4}
%e Triangle begins:
%e 1
%e 2
%e 3 4
%e 5 6 8
%e 7 10 9 12 16
%e 11 14 15 20 18 24 32
%e 13 22 21 25 28 30 27 40 36 48 64
%e 17 26 33 35 44 42 50 45 56 60 54 80 72 96 128
%e This corresponds to the following tetrangle:
%e 0
%e (1)
%e (2)(11)
%e (3)(21)(111)
%e (4)(31)(22)(211)(1111)
%e (5)(41)(32)(311)(221)(2111)(11111)
%t revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];
%t Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n],revlensort],{n,0,8}]
%Y Row lengths are A000041.
%Y Ignoring length gives A129129.
%Y Compositions under the same order are A296774 (triangle).
%Y The dual version (sum/length/lex) is A334433.
%Y The version for reversed partitions is A334435.
%Y The constructive version is A334439 (triangle).
%Y Lexicographically ordered reversed partitions are A026791.
%Y Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y Partitions in increasing-length colexicographic 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 order (sum/colex) are A211992.
%Y Graded Heinz numbers are given by A215366.
%Y Sorting partitions by Heinz number gives A296150.
%Y Cf. A056239, A066099, A124734, A185974, A228100, A228531, A333219, A334301, A334302, A334434, A334436, A334437.
%K nonn,tabf
%O 0,2
%A _Gus Wiseman_, May 03 2020
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