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%I A103921 #45 Sep 22 2023 08:45:36
%S A103921 0,1,1,1,1,2,1,1,2,1,2,1,1,2,2,2,2,2,1,1,2,2,1,2,3,1,2,2,2,1,1,2,2,2,
%T A103921 2,3,2,2,2,3,2,2,2,2,1,1,2,2,2,1,2,3,3,2,2,2,3,2,3,1,2,3,2,2,2,2,1,1,
%U A103921 2,2,2,2,2,3,3,2,2,3,1,2,3,3,3,3,2,2,3,2,3,2,2,3,2,2,2,2,1,1,2,2,2,2,1,2,3
%N A103921 Irregular triangle T(n,m) (n >= 0) read by rows: row n lists numbers of distinct parts of partitions of n in Abramowitz-Stegun order.
%C A103921 T(n, m) is the number of distinct parts of the m-th partition of n in Abramowitz-Stegun order; n >= 0, m = 1..p(n) = A000041(n).
%C A103921 The row length sequence of this table is p(n)=A000041(n) (number of partitions).
%C A103921 In order to count distinct parts of a partition consider the partition as a set instead of a multiset. E.g., n=6: read [1,1,1,3] as {1,3} and count the elements, here 2.
%C A103921 Rows are the same as the rows of A115623, but in reverse order.
%C A103921 From _Wolfdieter Lang_, Mar 17 2011: (Start)
%C A103921 The number of 1s in row number n, n >= 1, is tau(n)=A000005(n), the number of divisors of n.
%C A103921 For the proof read off the divisors d(n,j), j=1..tau(n), from row number n of table A027750, and translate them to the tau(n) partitions d(n,1)^(n/d(n,1)), d(n,2)^(n/d(n,2)),..., d(n,tau(n))^(n/d(n,tau(n))).
%C A103921 See a comment by _Giovanni Resta_ under A000005. (End)
%C A103921 From _Gus Wiseman_, May 20 2020: (Start)
%C A103921 The name is correct if integer partitions are read in reverse, so that the parts are weakly increasing. The non-reversed version is A334440.
%C A103921 Also the number of distinct parts of the n-th integer partition in lexicographic order (A193073).
%C A103921 Differs from the number of distinct parts in the n-th integer partition in (sum/length/revlex) order (A334439). For example, (6,2,2) has two distinct elements, while (1,4,5) has three.
%C A103921 (End)
%H A103921 Robert Price, <a href="/A103921/b103921.txt">Table of n, a(n) for n = 0..9295</a> (first 25 rows).
%H A103921 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 A103921 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, pp. 831-2.
%H A103921 Wolfdieter Lang, <a href="/A103921/a103921.pdf">First 10 rows.</a>
%H A103921 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A103921 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A103921 a(n) = A001221(A185974(n)). - _Gus Wiseman_, May 20 2020
%e A103921 Triangle starts:
%e A103921   0,
%e A103921   1,
%e A103921   1, 1,
%e A103921   1, 2, 1,
%e A103921   1, 2, 1, 2, 1,
%e A103921   1, 2, 2, 2, 2, 2, 1,
%e A103921   1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1,
%e A103921   1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1,
%e A103921   1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 2, 3, 2, 2, 2, 2, 1,
%e A103921   1, 2, 2, 2, 2, ...
%e A103921 a(5,4)=2 from the fourth partition of 5 in the mentioned order, i.e., (1^2,3), which has two distinct parts, namely 1 and 3.
%t A103921 Join@@Table[Length/@Union/@Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* _Gus Wiseman_, May 20 2020 *)
%Y A103921 Row sums are A000070.
%Y A103921 Row lengths are A000041.
%Y A103921 The lengths of these partitions are A036043.
%Y A103921 The maxima of these partitions are A049085.
%Y A103921 The version for non-reversed partitions is A334440.
%Y A103921 The version for colex instead of lex is (also) A334440.
%Y A103921 Lexicographically ordered reversed partitions are A026791.
%Y A103921 Reversed partitions in Abramowitz-Stegun order are A036036.
%Y A103921 Reverse-lexicographically ordered partitions are A080577.
%Y A103921 Compositions in Abramowitz-Stegun order are A124734.
%Y A103921 Cf. A001221, A036037, A112798, A115621, A115623, A185974, A193073, A228531, A334301, A334302, A334433, A334435, A334441.
%K A103921 nonn,tabf
%O A103921 0,6
%A A103921 _Wolfdieter Lang_, Mar 24 2005
%E A103921 Edited by _Franklin T. Adams-Watters_, May 29 2006

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