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Irregular table T(n,k) = maximal part of the k-th partition of n, when listed in Abramowitz-Stegun order (as in A036043).
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%I #50 Sep 22 2023 05:18:14

%S 0,1,2,1,3,2,1,4,3,2,2,1,5,4,3,3,2,2,1,6,5,4,3,4,3,2,3,2,2,1,7,6,5,4,

%T 5,4,3,3,4,3,2,3,2,2,1,8,7,6,5,4,6,5,4,4,3,5,4,3,3,2,4,3,2,3,2,2,1,9,

%U 8,7,6,5,7,6,5,4,5,4,3,6,5,4,4,3,3,5,4,3,3,2,4,3,2,3,2,2,1,10,9,8,7,6,5,8,7,6

%N Irregular table T(n,k) = maximal part of the k-th partition of n, when listed in Abramowitz-Stegun order (as in A036043).

%C a(0) = 0 by convention. - _Franklin T. Adams-Watters_, Jun 24 2014

%C Like A036043 this is important for calculating sequences defined over the numeric partitions, cf. A000041. For example, the triangular array A019575 can be calculated using A036042 and this sequence.

%C The row sums are A006128. - _Johannes W. Meijer_, Jun 21 2010

%C The name is correct if the partitions are read in reverse, so that the parts are weakly increasing. The version for non-reversed partitions is A334441. - _Gus Wiseman_, May 21 2020

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

%H Alois P. Heinz, <a href="/A049085/b049085.txt">Rows n = 0..26, flattened</a>

%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 Wolfdieter Lang, <a href="/A049085/a049085.txt">First 15 rows</a>.

%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>

%e Rows:

%e [0];

%e [1];

%e [2,1];

%e [3,2,1];

%e [4,3,2,2,1];

%e [5,4,3,3,2,2,1];

%e ...

%p with(combinat):

%p nmax:=9:

%p for n from 1 to nmax do

%p y(n):=numbpart(n):

%p P(n):=partition(n):

%p for k from 1 to y(n) do

%p B(k):=P(n)[k]

%p od:

%p for k from 1 to y(n) do

%p s:=0: j:=0:

%p while s<n do

%p j:=j+1: s:=s+B(k)[j]: Q(n,k):=j;

%p end do:

%p od:

%p od:

%p T:=0:

%p for n from 1 to nmax do

%p for j from 1 to numbpart(n) do

%p T:=T+1:

%p a(T):= Q(n,j)

%p od;

%p od:

%p 0, seq(a(n),n=1..T); # _Johannes W. Meijer_, Jun 21 2010

%t Table[If[n==0,{0},Max/@Sort[Reverse/@IntegerPartitions[n]]],{n,0,8}] (* _Gus Wiseman_, May 21 2020 *)

%o (PARI) A049085(n,k)=if(n,partitions(n)[k][1],0) \\ _M. F. Hasler_, Jun 06 2018

%Y Row lengths are A000041.

%Y Row sums are A006128.

%Y The length of the partition is A036043.

%Y The number of distinct elements of the partition is A103921.

%Y The Heinz number of the partition is A185974.

%Y The version ignoring length is A194546.

%Y The version for non-reversed partitions is A334441.

%Y Lexicographically ordered reversed partitions are A026791.

%Y Reversed partitions in Abramowitz-Stegun order are A036036.

%Y Reverse-lexicographically ordered partitions are A080577.

%Y Partitions in Abramowitz-Stegun order are A334301.

%Y Cf. A001221, A036037, A036042, A115623, A124734, A193073, A334302, A334433, A334438, A334439, A334440.

%K nice,nonn,tabf,look

%O 0,3

%A _Alford Arnold_

%E More terms from _Wolfdieter Lang_, Apr 28 2005

%E a(0) inserted by _Franklin T. Adams-Watters_, Jun 24 2014