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A334438 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

%I A334438 #17 Sep 22 2023 07:58:47
%S A334438 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 A334438 40,36,48,64,17,26,33,35,44,42,50,45,56,60,54,80,72,96,128,19,34,39,
%U A334438 55,49,52,66,70,63,75,88,84,100,90,81,112,120,108,160,144,192,256
%N A334438 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
%C A334438 First differs from A185974 shifted left once at a(76) = 99, A185974(75) = 98.
%C A334438 A permutation of the positive integers.
%C A334438 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 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.
%C A334438 As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.
%H A334438 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A334438 A001221(a(n)) = A103921(n).
%F A334438 A001222(a(n)) = A036043(n).
%e A334438 The sequence of terms together with their prime indices begins:
%e A334438     1: {}            32: {1,1,1,1,1}       50: {1,3,3}
%e A334438     2: {1}           13: {6}               45: {2,2,3}
%e A334438     3: {2}           22: {1,5}             56: {1,1,1,4}
%e A334438     4: {1,1}         21: {2,4}             60: {1,1,2,3}
%e A334438     5: {3}           25: {3,3}             54: {1,2,2,2}
%e A334438     6: {1,2}         28: {1,1,4}           80: {1,1,1,1,3}
%e A334438     8: {1,1,1}       30: {1,2,3}           72: {1,1,1,2,2}
%e A334438     7: {4}           27: {2,2,2}           96: {1,1,1,1,1,2}
%e A334438    10: {1,3}         40: {1,1,1,3}        128: {1,1,1,1,1,1,1}
%e A334438     9: {2,2}         36: {1,1,2,2}         19: {8}
%e A334438    12: {1,1,2}       48: {1,1,1,1,2}       34: {1,7}
%e A334438    16: {1,1,1,1}     64: {1,1,1,1,1,1}     39: {2,6}
%e A334438    11: {5}           17: {7}               55: {3,5}
%e A334438    14: {1,4}         26: {1,6}             49: {4,4}
%e A334438    15: {2,3}         33: {2,5}             52: {1,1,6}
%e A334438    20: {1,1,3}       35: {3,4}             66: {1,2,5}
%e A334438    18: {1,2,2}       44: {1,1,5}           70: {1,3,4}
%e A334438    24: {1,1,1,2}     42: {1,2,4}           63: {2,2,4}
%e A334438 Triangle begins:
%e A334438    1
%e A334438    2
%e A334438    3   4
%e A334438    5   6   8
%e A334438    7  10   9  12  16
%e A334438   11  14  15  20  18  24  32
%e A334438   13  22  21  25  28  30  27  40  36  48  64
%e A334438   17  26  33  35  44  42  50  45  56  60  54  80  72  96 128
%e A334438 This corresponds to the following tetrangle:
%e A334438                   0
%e A334438                  (1)
%e A334438                (2)(11)
%e A334438              (3)(21)(111)
%e A334438         (4)(31)(22)(211)(1111)
%e A334438   (5)(41)(32)(311)(221)(2111)(11111)
%t A334438 revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];
%t A334438 Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n],revlensort],{n,0,8}]
%Y A334438 Row lengths are A000041.
%Y A334438 Ignoring length gives A129129.
%Y A334438 Compositions under the same order are A296774 (triangle).
%Y A334438 The dual version (sum/length/lex) is A334433.
%Y A334438 The version for reversed partitions is A334435.
%Y A334438 The constructive version is A334439 (triangle).
%Y A334438 Lexicographically ordered reversed partitions are A026791.
%Y A334438 Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y A334438 Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
%Y A334438 Reverse-lexicographically ordered partitions are A080577.
%Y A334438 Sorting reversed partitions by Heinz number gives A112798.
%Y A334438 Graded lexicographically ordered partitions are A193073.
%Y A334438 Partitions in colexicographic order (sum/colex) are A211992.
%Y A334438 Graded Heinz numbers are given by A215366.
%Y A334438 Sorting partitions by Heinz number gives A296150.
%Y A334438 Cf. A056239, A066099, A124734, A185974, A228100, A228531, A333219, A334301, A334302, A334434, A334436, A334437.
%K A334438 nonn,tabf
%O A334438 0,2
%A A334438 _Gus Wiseman_, May 03 2020