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%I #53 Jan 12 2024 22:45:11
%S 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 40,36,48,64,17,26,33,35,44,42,50,45,56,60,54,80,72,96,128,19,34,39,
%U 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 Partitions in Abramowitz-Stegun order A036036 mapped one-to-one to positive integers.
%C First differs from A334438 (shifted left once) at a(75) = 98, A334438(76) = 99. - _Gus Wiseman_, May 20 2020
%C 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 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 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 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 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 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>.
%F 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 From _Gus Wiseman_, May 21 2020: (Start)
%F A001221(a(n)) = A103921(n).
%F A001222(a(n)) = A036043(n).
%F A056239(a(n)) = A036042(n).
%F A061395(a(n)) = A049085(n).
%F (End)
%e 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 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 From _Gus Wiseman_, May 20 2020: (Start)
%e Triangle begins:
%e 1
%e 2
%e 3 4
%e 5 6 8
%e 7 10 9 12 16
%e 11 14 15 20 18 24 32
%e 13 22 21 25 28 30 27 40 36 48 64
%e 17 26 33 35 44 42 50 45 56 60 54 80 72 96 128
%e As a triangle of reversed partitions we have:
%e 0
%e (1)
%e (2)(11)
%e (3)(12)(111)
%e (4)(13)(22)(112)(1111)
%e (5)(14)(23)(113)(122)(1112)(11111)
%e (6)(15)(24)(33)(114)(123)(222)(1113)(1122)(11112)(111111)
%e (End)
%t Join@@Table[Times@@Prime/@#&/@Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* _Gus Wiseman_, May 21 2020 *)
%o From _M. F. Hasler_, Jan 07 2024: (Start)
%o (PARI) A185974_row(n)=[vecprod([prime(i)|i<-p])|p<-partitions(n)] \\ below a helper function:
%o index_of_partition(n)={for(r=0, oo, my(c = numbpart(r)); n >= c || return([r,n+1]); n -= c)}
%o /* 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 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(n, k=index_of_partition(n))=A185974_row(iferr(k[1], E, k=[k,k]; n))[k[2]] \\ (End)
%Y Row lengths are A000041.
%Y The constructive version is A036036.
%Y Also Heinz numbers of the partitions in A036037.
%Y The generalization to compositions is A124734.
%Y The version for non-reversed partitions is A334433.
%Y The non-reversed length-insensitive version is A334434.
%Y The opposite version (sum/length/revlex) is A334435.
%Y Ignoring length gives A334437.
%Y Sorting reversed partitions by Heinz number gives A112798.
%Y Partitions in lexicographic order are A193073.
%Y Partitions in colexicographic order are A211992.
%Y Graded Heinz numbers are A215366.
%Y Cf. A026791, A036043, A056239, A080577, A228531, A296150, A334301, A334302, A334436, A334438, A334439.
%K nonn,easy,tabf
%O 0,2
%A _Wolfdieter Lang_, Feb 10 2011
%E Examples edited by _M. F. Hasler_, Jan 07 2024
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