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A103921
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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.
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26
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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, 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, 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
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OFFSET
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0,6
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COMMENTS
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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).
The row length sequence of this table is p(n)=A000041(n) (number of partitions).
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.
Rows are the same as the rows of A115623, but in reverse order.
From Wolfdieter Lang, Mar 17 2011: (Start)
The number of 1s in row number n, n >= 1, is tau(n)=A000005(n), the number of divisors of n.
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))).
See a comment by Giovanni Resta under A000005. (End)
From Gus Wiseman, May 20 2020: (Start)
The name is correct if integer partitions are read in reverse, so that the parts are weakly increasing. The non-reversed version is A334440.
Also the number of distinct parts of the n-th integer partition in lexicographic order (A193073).
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.
(End)
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LINKS
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Robert Price, Table of n, a(n) for n = 0..9295 (first 25 rows).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wolfdieter Lang, First 10 rows.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order
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FORMULA
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a(n) = A001221(A185974(n)). - Gus Wiseman, May 20 2020
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EXAMPLE
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Triangle starts:
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, 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, 2, 2, 2, 2, ...
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.
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MATHEMATICA
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Join@@Table[Length/@Union/@Sort[Reverse/@IntegerPartitions[n]], {n, 0, 8}] (* Gus Wiseman, May 20 2020 *)
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CROSSREFS
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Row sums are A000070.
Row lengths are A000041.
The lengths of these partitions are A036043.
The maxima of these partitions are A049085.
The version for non-reversed partitions is A334440.
The version for colex instead of lex is (also) A334440.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order are A036036.
Reverse-lexicographically ordered partitions are A080577.
Compositions in Abramowitz-Stegun order are A124734.
Cf. A001221, A036037, A112798, A115621, A115623, A185974, A193073, A228531, A334301, A334302, A334433, A334435, A334441.
Sequence in context: A007421 A239228 A346080 * A115623 A279044 A134265
Adjacent sequences: A103918 A103919 A103920 * A103922 A103923 A103924
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KEYWORD
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nonn,tabf
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AUTHOR
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Wolfdieter Lang, Mar 24 2005
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EXTENSIONS
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Edited by Franklin T. Adams-Watters, May 29 2006
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STATUS
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approved
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