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A334435 Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

%I A334435 #20 Sep 22 2023 07:53:08
%S A334435 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 A334435 36,40,48,64,17,35,33,26,45,50,42,44,54,60,56,72,80,96,128,19,49,55,
%U A334435 39,34,75,63,70,66,52,81,90,100,84,88,108,120,112,144,160,192,256
%N A334435 Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
%C A334435 First differs from A334433 at a(75) = 99, A334433(75) = 98.
%C A334435 First differs from A334436 at a(22) = 22, A334436(22) = 27.
%C A334435 A permutation of the positive integers.
%C A334435 Reversed integer partitions are finite weakly increasing sequences of positive integers.
%C A334435 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 A334435 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 A334435 As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.
%H A334435 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A334435 A001222(a(n)) = A036043(n).
%e A334435 The sequence of terms together with their prime indices begins:
%e A334435     1: {}            32: {1,1,1,1,1}       42: {1,2,4}
%e A334435     2: {1}           13: {6}               44: {1,1,5}
%e A334435     3: {2}           25: {3,3}             54: {1,2,2,2}
%e A334435     4: {1,1}         21: {2,4}             60: {1,1,2,3}
%e A334435     5: {3}           22: {1,5}             56: {1,1,1,4}
%e A334435     6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
%e A334435     8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
%e A334435     7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
%e A334435     9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
%e A334435    10: {1,3}         40: {1,1,1,3}         19: {8}
%e A334435    12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
%e A334435    16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
%e A334435    11: {5}           17: {7}               39: {2,6}
%e A334435    15: {2,3}         35: {3,4}             34: {1,7}
%e A334435    14: {1,4}         33: {2,5}             75: {2,3,3}
%e A334435    18: {1,2,2}       26: {1,6}             63: {2,2,4}
%e A334435    20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
%e A334435    24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
%e A334435 Triangle begins:
%e A334435    1
%e A334435    2
%e A334435    3   4
%e A334435    5   6   8
%e A334435    7   9  10  12  16
%e A334435   11  15  14  18  20  24  32
%e A334435   13  25  21  22  27  30  28  36  40  48  64
%e A334435   17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
%e A334435 This corresponds to the following tetrangle:
%e A334435                   0
%e A334435                  (1)
%e A334435                (2)(11)
%e A334435              (3)(12)(111)
%e A334435         (4)(22)(13)(112)(1111)
%e A334435   (5)(23)(14)(122)(113)(1112)(11111)
%t A334435 revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];
%t A334435 Table[Times@@Prime/@#&/@Sort[Sort/@IntegerPartitions[n],revlensort],{n,0,8}]
%Y A334435 Row lengths are A000041.
%Y A334435 The dual version (sum/length/lex) is A185974.
%Y A334435 Compositions under the same order are A296774 (triangle).
%Y A334435 The constructive version is A334302.
%Y A334435 Ignoring length gives A334436.
%Y A334435 The version for non-reversed partitions is A334438.
%Y A334435 Partitions in this order (sum/length/revlex) are A334439.
%Y A334435 Lexicographically ordered reversed partitions are A026791.
%Y A334435 Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y A334435 Partitions in increasing-length colex order (sum/length/colex) are A036037.
%Y A334435 Reverse-lexicographically ordered partitions are A080577.
%Y A334435 Sorting reversed partitions by Heinz number gives A112798.
%Y A334435 Graded lexicographically ordered partitions are A193073.
%Y A334435 Partitions in colexicographic (sum/colex) order are A211992.
%Y A334435 Graded Heinz numbers are given by A215366.
%Y A334435 Sorting partitions by Heinz number gives A296150.
%Y A334435 Cf. A056239, A124734, A129129, A228100, A228531, A333219, A333220, A334301, A334433, A334434, A334437.
%K A334435 nonn,tabf
%O A334435 0,2
%A A334435 _Gus Wiseman_, May 02 2020