cp's OEIS Frontend

A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

%I A334433 #21 May 28 2020 05:03:13
%S A334433 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 A334433 36,40,48,64,17,35,33,26,45,50,42,44,54,60,56,72,80,96,128,19,49,55,
%U A334433 39,34,75,63,70,66,52,81,90,100,84,88,108,120,112,144,160,192,256
%N A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.
%C A334433 First differs from A334435 at a(75) = 99, A334435(75) = 98.
%C A334433 A permutation of the positive integers.
%C A334433 This is the Abramowitz-Stegun ordering of integer partitions when the parts are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A185974.
%C A334433 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 A334433 As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.
%H A334433 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A334433 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A334433 A001222(a(n)) = A036043(n).
%e A334433 The sequence of terms together with their prime indices begins:
%e A334433     1: {}            32: {1,1,1,1,1}       42: {1,2,4}
%e A334433     2: {1}           13: {6}               44: {1,1,5}
%e A334433     3: {2}           25: {3,3}             54: {1,2,2,2}
%e A334433     4: {1,1}         21: {2,4}             60: {1,1,2,3}
%e A334433     5: {3}           22: {1,5}             56: {1,1,1,4}
%e A334433     6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
%e A334433     8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
%e A334433     7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
%e A334433     9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
%e A334433    10: {1,3}         40: {1,1,1,3}         19: {8}
%e A334433    12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
%e A334433    16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
%e A334433    11: {5}           17: {7}               39: {2,6}
%e A334433    15: {2,3}         35: {3,4}             34: {1,7}
%e A334433    14: {1,4}         33: {2,5}             75: {2,3,3}
%e A334433    18: {1,2,2}       26: {1,6}             63: {2,2,4}
%e A334433    20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
%e A334433    24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
%e A334433 Triangle begins:
%e A334433    1
%e A334433    2
%e A334433    3   4
%e A334433    5   6   8
%e A334433    7   9  10  12  16
%e A334433   11  15  14  18  20  24  32
%e A334433   13  25  21  22  27  30  28  36  40  48  64
%e A334433   17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
%e A334433 This corresponds to the tetrangle:
%e A334433                   0
%e A334433                  (1)
%e A334433                (2)(11)
%e A334433              (3)(21)(111)
%e A334433         (4)(22)(31)(211)(1111)
%e A334433   (5)(32)(41)(221)(311)(2111)(11111)
%t A334433 Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n]],{n,0,8}]
%Y A334433 Row lengths are A000041.
%Y A334433 Compositions under the same order are A124734 (triangle).
%Y A334433 The version for reversed (weakly increasing) partitions is A185974.
%Y A334433 The constructive version is A334301.
%Y A334433 Ignoring length gives A334434, or A334437 for reversed partitions.
%Y A334433 The dual version (sum/length/revlex) is A334438.
%Y A334433 Lexicographically ordered reversed partitions are A026791.
%Y A334433 Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y A334433 Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
%Y A334433 Graded reverse-lexicographically ordered partitions are A080577.
%Y A334433 Sorting reversed partitions by Heinz number gives A112798.
%Y A334433 Graded lexicographically ordered partitions are A193073.
%Y A334433 Graded Heinz numbers are A215366.
%Y A334433 Sorting partitions by Heinz number gives A296150.
%Y A334433 Partitions in increasing-length reverse-lexicographic order (sum/length/revlex) are A334439 (not A036037).
%Y A334433 Cf. A026791, A036043, A048793, A056239, A129129, A211992, A228351, A228531, A334302, A334435, A334436.
%K A334433 nonn,tabf
%O A334433 0,2
%A A334433 _Gus Wiseman_, Apr 30 2020