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%I #12 May 03 2020 23:06:16
%S 1,2,4,3,8,6,5,16,12,9,10,7,32,24,18,20,15,14,11,64,48,36,27,40,30,25,
%T 28,21,22,13,128,96,72,54,80,60,45,50,56,42,35,44,33,26,17,256,192,
%U 144,108,81,160,120,90,100,75,112,84,63,70,49,88,66,55,52,39,34,19
%N Heinz number of the n-th integer partition in graded lexicographic order.
%C A permutation of the positive integers.
%C This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).
%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},{4,3},{8,6,5},...}, so offset is 0.
%H OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A001222(a(n)) appears to be A049085(n).
%e The sequence of terms together with their prime indices begins:
%e 1: {} 11: {5} 45: {2,2,3}
%e 2: {1} 64: {1,1,1,1,1,1} 50: {1,3,3}
%e 4: {1,1} 48: {1,1,1,1,2} 56: {1,1,1,4}
%e 3: {2} 36: {1,1,2,2} 42: {1,2,4}
%e 8: {1,1,1} 27: {2,2,2} 35: {3,4}
%e 6: {1,2} 40: {1,1,1,3} 44: {1,1,5}
%e 5: {3} 30: {1,2,3} 33: {2,5}
%e 16: {1,1,1,1} 25: {3,3} 26: {1,6}
%e 12: {1,1,2} 28: {1,1,4} 17: {7}
%e 9: {2,2} 21: {2,4} 256: {1,1,1,1,1,1,1,1}
%e 10: {1,3} 22: {1,5} 192: {1,1,1,1,1,1,2}
%e 7: {4} 13: {6} 144: {1,1,1,1,2,2}
%e 32: {1,1,1,1,1} 128: {1,1,1,1,1,1,1} 108: {1,1,2,2,2}
%e 24: {1,1,1,2} 96: {1,1,1,1,1,2} 81: {2,2,2,2}
%e 18: {1,2,2} 72: {1,1,1,2,2} 160: {1,1,1,1,1,3}
%e 20: {1,1,3} 54: {1,2,2,2} 120: {1,1,1,2,3}
%e 15: {2,3} 80: {1,1,1,1,3} 90: {1,2,2,3}
%e 14: {1,4} 60: {1,1,2,3} 100: {1,1,3,3}
%e Triangle begins:
%e 1
%e 2
%e 4 3
%e 8 6 5
%e 16 12 9 10 7
%e 32 24 18 20 15 14 11
%e 64 48 36 27 40 30 25 28 21 22 13
%e 128 96 72 54 80 60 45 50 56 42 35 44 33 26 17
%e This corresponds to the tetrangle:
%e 0
%e (1)
%e (11)(2)
%e (111)(21)(3)
%e (1111)(211)(22)(31)(4)
%e (11111)(2111)(221)(311)(32)(41)(5)
%t lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
%t Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n],lexsort],{n,0,8}]
%t - or -
%t Join@@Table[Times@@Prime/@#&/@Reverse[IntegerPartitions[n]],{n,0,8}]
%Y Row lengths are A000041.
%Y The dual version (sum/revlex) is A129129.
%Y The constructive version is A193073.
%Y Compositions under the same order are A228351.
%Y The length-sensitive version is A334433.
%Y The version for reversed (weakly increasing) partitions is A334437.
%Y Lexicographically ordered reversed partitions are A026791.
%Y Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
%Y Reverse-lexicographically ordered partitions are A080577.
%Y Sorting reversed partitions by Heinz number gives A112798.
%Y Graded Heinz numbers are A215366.
%Y Sorting partitions by Heinz number gives A296150.
%Y Cf. A036037, A049085, A056239, A066099, A185974, A211992, A228100, A228531, A333219, A334301, A334302, A334435, A334436, A334438, A334439.
%K nonn,tabf
%O 0,2
%A _Gus Wiseman_, May 01 2020
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