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A049085 Irregular table T(n,k) = maximal part of the k-th partition of n, when listed in Abramowitz-Stegun order (as in A036043).

%I A049085 #50 Sep 22 2023 05:18:14
%S A049085 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 A049085 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 A049085 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 A049085 Irregular table T(n,k) = maximal part of the k-th partition of n, when listed in Abramowitz-Stegun order (as in A036043).
%C A049085 a(0) = 0 by convention. - _Franklin T. Adams-Watters_, Jun 24 2014
%C A049085 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 A049085 The row sums are A006128. - _Johannes W. Meijer_, Jun 21 2010
%C A049085 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 A049085 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 A049085 Alois P. Heinz, <a href="/A049085/b049085.txt">Rows n = 0..26, flattened</a>
%H A049085 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 A049085 Wolfdieter Lang, <a href="/A049085/a049085.txt">First 15 rows</a>.
%H A049085 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A049085 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A049085 Rows:
%e A049085   [0];
%e A049085   [1];
%e A049085   [2,1];
%e A049085   [3,2,1];
%e A049085   [4,3,2,2,1];
%e A049085   [5,4,3,3,2,2,1];
%e A049085   ...
%p A049085 with(combinat):
%p A049085 nmax:=9:
%p A049085 for n from 1 to nmax do
%p A049085    y(n):=numbpart(n):
%p A049085    P(n):=partition(n):
%p A049085    for k from 1 to y(n) do
%p A049085       B(k):=P(n)[k]
%p A049085    od:
%p A049085    for k from 1 to y(n) do
%p A049085       s:=0: j:=0:
%p A049085       while s<n do
%p A049085          j:=j+1: s:=s+B(k)[j]: Q(n,k):=j;
%p A049085       end do:
%p A049085    od:
%p A049085 od:
%p A049085 T:=0:
%p A049085 for n from 1 to nmax do
%p A049085    for j from 1 to numbpart(n) do
%p A049085       T:=T+1:
%p A049085       a(T):= Q(n,j)
%p A049085     od;
%p A049085 od:
%p A049085 0, seq(a(n),n=1..T); # _Johannes W. Meijer_, Jun 21 2010
%t A049085 Table[If[n==0,{0},Max/@Sort[Reverse/@IntegerPartitions[n]]],{n,0,8}] (* _Gus Wiseman_, May 21 2020 *)
%o A049085 (PARI) A049085(n,k)=if(n,partitions(n)[k][1],0) \\ _M. F. Hasler_, Jun 06 2018
%Y A049085 Row lengths are A000041.
%Y A049085 Row sums are A006128.
%Y A049085 The length of the partition is A036043.
%Y A049085 The number of distinct elements of the partition is A103921.
%Y A049085 The Heinz number of the partition is A185974.
%Y A049085 The version ignoring length is A194546.
%Y A049085 The version for non-reversed partitions is A334441.
%Y A049085 Lexicographically ordered reversed partitions are A026791.
%Y A049085 Reversed partitions in Abramowitz-Stegun order are A036036.
%Y A049085 Reverse-lexicographically ordered partitions are A080577.
%Y A049085 Partitions in Abramowitz-Stegun order are A334301.
%Y A049085 Cf. A001221, A036037, A036042, A115623, A124734, A193073, A334302, A334433, A334438, A334439, A334440.
%K A049085 nice,nonn,tabf,look
%O A049085 0,3
%A A049085 _Alford Arnold_
%E A049085 More terms from _Wolfdieter Lang_, Apr 28 2005
%E A049085 a(0) inserted by _Franklin T. Adams-Watters_, Jun 24 2014