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A334436 Heinz numbers of all reversed integer partitions sorted first by sum and then reverse-lexicographically.

%I A334436 #14 Sep 22 2023 08:43:40
%S A334436 1,2,3,4,5,6,8,7,9,10,12,16,11,15,14,18,20,24,32,13,25,21,27,22,30,28,
%T A334436 36,40,48,64,17,35,33,45,26,50,42,54,44,60,56,72,80,96,128,19,49,55,
%U A334436 39,75,63,81,34,70,66,90,52,100,84,108,88,120,112,144,160,192,256
%N A334436 Heinz numbers of all reversed integer partitions sorted first by sum and then reverse-lexicographically.
%C A334436 First differs from A334435 at a(22) = 27, A334435(22) = 22.
%C A334436 A permutation of the positive integers.
%C A334436 Reversed integer partitions are finite weakly increasing sequences of positive integers. For non-reversed partitions, see A129129 and A228531.
%C A334436 This is the so-called "Mathematica" order (A080577).
%C A334436 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.
%H A334436 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A334436 A001222(a(n)) = A333486(n).
%e A334436 The sequence of terms together with their prime indices begins:
%e A334436     1: {}            32: {1,1,1,1,1}       42: {1,2,4}
%e A334436     2: {1}           13: {6}               54: {1,2,2,2}
%e A334436     3: {2}           25: {3,3}             44: {1,1,5}
%e A334436     4: {1,1}         21: {2,4}             60: {1,1,2,3}
%e A334436     5: {3}           27: {2,2,2}           56: {1,1,1,4}
%e A334436     6: {1,2}         22: {1,5}             72: {1,1,1,2,2}
%e A334436     8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
%e A334436     7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
%e A334436     9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
%e A334436    10: {1,3}         40: {1,1,1,3}         19: {8}
%e A334436    12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
%e A334436    16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
%e A334436    11: {5}           17: {7}               39: {2,6}
%e A334436    15: {2,3}         35: {3,4}             75: {2,3,3}
%e A334436    14: {1,4}         33: {2,5}             63: {2,2,4}
%e A334436    18: {1,2,2}       45: {2,2,3}           81: {2,2,2,2}
%e A334436    20: {1,1,3}       26: {1,6}             34: {1,7}
%e A334436    24: {1,1,1,2}     50: {1,3,3}           70: {1,3,4}
%e A334436 Triangle begins:
%e A334436    1
%e A334436    2
%e A334436    3   4
%e A334436    5   6   8
%e A334436    7   9  10  12  16
%e A334436   11  15  14  18  20  24  32
%e A334436   13  25  21  27  22  30  28  36  40  48  64
%e A334436   17  35  33  45  26  50  42  54  44  60  56  72  80  96 128
%e A334436 This corresponds to the following tetrangle:
%e A334436                   0
%e A334436                  (1)
%e A334436                (2)(11)
%e A334436              (3)(12)(111)
%e A334436         (4)(22)(13)(112)(1111)
%e A334436   (5)(23)(14)(122)(113)(1112)(11111)
%t A334436 lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
%t A334436 Table[Times@@Prime/@#&/@Reverse[Sort[Sort/@IntegerPartitions[n],lexsort]],{n,0,8}]
%Y A334436 Row lengths are A000041.
%Y A334436 Compositions under the same order are A066099 (triangle).
%Y A334436 The version for non-reversed partitions is A129129.
%Y A334436 The constructive version is A228531.
%Y A334436 The lengths of these partitions are A333486.
%Y A334436 The length-sensitive version is A334435.
%Y A334436 The dual version (sum/lex) is A334437.
%Y A334436 Lexicographically ordered reversed partitions are A026791.
%Y A334436 Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y A334436 Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
%Y A334436 Reverse-lexicographically ordered partitions are A080577.
%Y A334436 Sorting reversed partitions by Heinz number gives A112798.
%Y A334436 Graded lexicographically ordered partitions are A193073.
%Y A334436 Partitions in colexicographic order (sum/colex) are A211992.
%Y A334436 Graded Heinz numbers are A215366.
%Y A334436 Sorting partitions by Heinz number gives A296150.
%Y A334436 Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
%Y A334436 Cf. A056239, A124734, A185974, A228100, A333219, A334301, A334302, A334433, A334434, A334438.
%K A334436 nonn,tabf
%O A334436 0,2
%A A334436 _Gus Wiseman_, May 02 2020