This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334437 #14 Sep 22 2023 08:56:41
%S A334437 1,2,4,3,8,6,5,16,12,10,9,7,32,24,20,18,14,15,11,64,48,40,36,28,30,22,
%T A334437 27,21,25,13,128,96,80,72,56,60,44,54,42,50,26,45,33,35,17,256,192,
%U A334437 160,144,112,120,88,108,84,100,52,90,66,70,34,81,63,75,39,55,49,19
%N A334437 Heinz number of the n-th reversed integer partition in graded lexicographical order.
%C A334437 A permutation of the positive integers.
%C A334437 Reversed integer partitions are finite weakly increasing sequences of positive integers. The non-reversed version is A334434.
%C A334437 This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).
%C A334437 The Heinz number of a reversed integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and reversed partitions.
%C A334437 Also Heinz numbers of partitions in colexicographic order (cf. A211992).
%C A334437 As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, so offset is 0.
%H A334437 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A334437 A001222(a(n)) = A193173(n).
%e A334437 The sequence of terms together with their prime indices begins:
%e A334437 1: {} 11: {5} 44: {1,1,5}
%e A334437 2: {1} 64: {1,1,1,1,1,1} 54: {1,2,2,2}
%e A334437 4: {1,1} 48: {1,1,1,1,2} 42: {1,2,4}
%e A334437 3: {2} 40: {1,1,1,3} 50: {1,3,3}
%e A334437 8: {1,1,1} 36: {1,1,2,2} 26: {1,6}
%e A334437 6: {1,2} 28: {1,1,4} 45: {2,2,3}
%e A334437 5: {3} 30: {1,2,3} 33: {2,5}
%e A334437 16: {1,1,1,1} 22: {1,5} 35: {3,4}
%e A334437 12: {1,1,2} 27: {2,2,2} 17: {7}
%e A334437 10: {1,3} 21: {2,4} 256: {1,1,1,1,1,1,1,1}
%e A334437 9: {2,2} 25: {3,3} 192: {1,1,1,1,1,1,2}
%e A334437 7: {4} 13: {6} 160: {1,1,1,1,1,3}
%e A334437 32: {1,1,1,1,1} 128: {1,1,1,1,1,1,1} 144: {1,1,1,1,2,2}
%e A334437 24: {1,1,1,2} 96: {1,1,1,1,1,2} 112: {1,1,1,1,4}
%e A334437 20: {1,1,3} 80: {1,1,1,1,3} 120: {1,1,1,2,3}
%e A334437 18: {1,2,2} 72: {1,1,1,2,2} 88: {1,1,1,5}
%e A334437 14: {1,4} 56: {1,1,1,4} 108: {1,1,2,2,2}
%e A334437 15: {2,3} 60: {1,1,2,3} 84: {1,1,2,4}
%e A334437 Triangle begins:
%e A334437 1
%e A334437 2
%e A334437 4 3
%e A334437 8 6 5
%e A334437 16 12 10 9 7
%e A334437 32 24 20 18 14 15 11
%e A334437 64 48 40 36 28 30 22 27 21 25 13
%e A334437 128 96 80 72 56 60 44 54 42 50 26 45 33 35 17
%e A334437 This corresponds to the following tetrangle:
%e A334437 0
%e A334437 (1)
%e A334437 (11)(2)
%e A334437 (111)(12)(3)
%e A334437 (1111)(112)(13)(22)(4)
%e A334437 (11111)(1112)(113)(122)(14)(23)(5)
%t A334437 lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
%t A334437 Table[Times@@Prime/@#&/@Sort[Sort/@IntegerPartitions[n],lexsort],{n,0,8}]
%Y A334437 Row lengths are A000041.
%Y A334437 The constructive version is A026791 (triangle).
%Y A334437 The length-sensitive version is A185974.
%Y A334437 Compositions under the same order are A228351 (triangle).
%Y A334437 The version for non-reversed partitions is A334434.
%Y A334437 The dual version (sum/revlex) is A334436.
%Y A334437 Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y A334437 Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
%Y A334437 Graded reverse-lexicographically ordered partitions are A080577.
%Y A334437 Sorting reversed partitions by Heinz number gives A112798.
%Y A334437 Graded lexicographically ordered partitions are A193073.
%Y A334437 Partitions in colexicographic order (sum/colex) are A211992.
%Y A334437 Graded Heinz numbers are given by A215366.
%Y A334437 Sorting partitions by Heinz number gives A296150.
%Y A334437 Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
%Y A334437 Cf. A056239, A066099, A129129, A228531, A333219, A333220, A334301, A334302, A334433, A334435, A334438.
%K A334437 nonn,tabf
%O A334437 0,2
%A A334437 _Gus Wiseman_, May 03 2020