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A185974 Partitions in Abramowitz-Stegun order A036036 mapped one-to-one to positive integers.

%I A185974 #53 Jan 12 2024 22:45:11
%S A185974 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 A185974 40,36,48,64,17,26,33,35,44,42,50,45,56,60,54,80,72,96,128,19,34,39,
%U A185974 55,49,52,66,70,63,75,88,84,100,90,81,112,120,108,160,144,192,256,23,38,51,65,77,68,78,110,98,99,105,125,104,132,140,126,150,135,176,168,200,180,162,224,240,216,320,288,384,512,29,46,57,85,91,121,76,102,130,154,117,165,147,175,136,156,220,196,198,210,250,189,225,208,264,280,252,300,270,243,352,336,400,360,324,448,480,432,640,576,768,1024
%N A185974 Partitions in Abramowitz-Stegun order A036036 mapped one-to-one to positive integers.
%C A185974 First differs from A334438 (shifted left once) at a(75) = 98, A334438(76) = 99. - _Gus Wiseman_, May 20 2020
%C A185974 This mapping of the set of all partitions of N >= 0 to {1, 2, 3, ...} (set of natural numbers) is one to one (bijective). The empty partition for N = 0 maps to 1.
%C A185974 A129129 seems to be analogous, except that the partition ordering A080577 is used. This ordering, however, does not care about the number of parts: e.g., 1^2,4 = 4,1^2 comes before 3^2, so a(23)=28 and a(22)=25 are interchanged.
%C A185974 Also Heinz numbers of all reversed integer partitions (finite weakly increasing sequences of positive integers), sorted first by sum, then by length, and finally lexicographically, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The version for non-reversed partitions is A334433. - _Gus Wiseman_, May 20 2020
%H A185974 M. F. Hasler, <a href="/A185974/b185974.txt">Table of n, a(n) for n = 0..9295</a> (up to partitions of 25), Jan 07 2024
%H A185974 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A185974 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A185974 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>.
%F A185974 a(n) = Product_{j=1..N(n)} p(j)^e(j), with p(j):=A000040(j) (j-th prime), and the exponent e(j) >= 0 of the part j in the n-th partition written in Abramowitz-Stegun (A-St) order, indicated in A036036. Note that j^0 is not 1 but has to be omitted in the partition. N(n) is the index (argument) of the smallest A026905-number greater than or equal to n (the index of the A026905-ceiling of n).
%F A185974 From _Gus Wiseman_, May 21 2020: (Start)
%F A185974 A001221(a(n)) = A103921(n).
%F A185974 A001222(a(n)) = A036043(n).
%F A185974 A056239(a(n)) = A036042(n).
%F A185974 A061395(a(n)) = A049085(n).
%F A185974 (End)
%e A185974 a(22) = 25 = prime(3)^2 because the 22nd partition in A-St order is the 2-part partition (3,3) of N = 6, because A026905(5) = 18 < 22 <= A026905(6) = 29.
%e A185974 a(23) = 28 = prime(1)^2*prime(4) corresponds to the partition 1+1+4 = 4+1+1 with three parts, also of N = 6.
%e A185974 From _Gus Wiseman_, May 20 2020: (Start)
%e A185974 Triangle begins:
%e A185974    1
%e A185974    2
%e A185974    3   4
%e A185974    5   6   8
%e A185974    7  10   9  12  16
%e A185974   11  14  15  20  18  24  32
%e A185974   13  22  21  25  28  30  27  40  36  48  64
%e A185974   17  26  33  35  44  42  50  45  56  60  54  80  72  96 128
%e A185974 As a triangle of reversed partitions we have:
%e A185974                              0
%e A185974                             (1)
%e A185974                           (2)(11)
%e A185974                         (3)(12)(111)
%e A185974                    (4)(13)(22)(112)(1111)
%e A185974              (5)(14)(23)(113)(122)(1112)(11111)
%e A185974   (6)(15)(24)(33)(114)(123)(222)(1113)(1122)(11112)(111111)
%e A185974 (End)
%t A185974 Join@@Table[Times@@Prime/@#&/@Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* _Gus Wiseman_, May 21 2020 *)
%o A185974 From _M. F. Hasler_, Jan 07 2024: (Start)
%o A185974 (PARI) A185974_row(n)=[vecprod([prime(i)|i<-p])|p<-partitions(n)] \\ below a helper function:
%o A185974 index_of_partition(n)={for(r=0, oo, my(c = numbpart(r)); n >= c || return([r,n+1]); n -= c)}
%o A185974 /* A185974(n,k), 1 <= k <= A000041(n), gives the k-th partition of n >= 0; if k is omitted, A185974(n) return the term of index n of the flattened sequence a(n >= 0).
%o A185974   This function is used in other sequences (such as A122172) which need to access the n-th partition as listed in A-S order. */
%o A185974 A185974(n, k=index_of_partition(n))=A185974_row(iferr(k[1], E, k=[k,k]; n))[k[2]] \\ (End)
%Y A185974 Row lengths are A000041.
%Y A185974 The constructive version is A036036.
%Y A185974 Also Heinz numbers of the partitions in A036037.
%Y A185974 The generalization to compositions is A124734.
%Y A185974 The version for non-reversed partitions is A334433.
%Y A185974 The non-reversed length-insensitive version is A334434.
%Y A185974 The opposite version (sum/length/revlex) is A334435.
%Y A185974 Ignoring length gives A334437.
%Y A185974 Sorting reversed partitions by Heinz number gives A112798.
%Y A185974 Partitions in lexicographic order are A193073.
%Y A185974 Partitions in colexicographic order are A211992.
%Y A185974 Graded Heinz numbers are A215366.
%Y A185974 Cf. A026791, A036043, A056239, A080577, A228531, A296150, A334301, A334302, A334436, A334438, A334439.
%K A185974 nonn,easy,tabf
%O A185974 0,2
%A A185974 _Wolfdieter Lang_, Feb 10 2011
%E A185974 Examples edited by _M. F. Hasler_, Jan 07 2024