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A049085
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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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27
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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, 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, 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
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OFFSET
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0,3
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COMMENTS
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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.
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
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REFERENCES
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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.
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LINKS
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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].
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EXAMPLE
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Rows:
[0];
[1];
[2,1];
[3,2,1];
[4,3,2,2,1];
[5,4,3,3,2,2,1];
...
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MAPLE
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with(combinat):
nmax:=9:
for n from 1 to nmax do
y(n):=numbpart(n):
P(n):=partition(n):
for k from 1 to y(n) do
B(k):=P(n)[k]
od:
for k from 1 to y(n) do
s:=0: j:=0:
while s<n do
j:=j+1: s:=s+B(k)[j]: Q(n, k):=j;
end do:
od:
od:
T:=0:
for n from 1 to nmax do
for j from 1 to numbpart(n) do
T:=T+1:
a(T):= Q(n, j)
od;
od:
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MATHEMATICA
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Table[If[n==0, {0}, Max/@Sort[Reverse/@IntegerPartitions[n]]], {n, 0, 8}] (* Gus Wiseman, May 21 2020 *)
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PROG
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CROSSREFS
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The length of the partition is A036043.
The number of distinct elements of the partition is A103921.
The Heinz number of the partition is A185974.
The version ignoring length is A194546.
The version for non-reversed partitions is A334441.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order are A036036.
Reverse-lexicographically ordered partitions are A080577.
Partitions in Abramowitz-Stegun order are A334301.
Cf. A001221, A036037, A036042, A115623, A124734, A193073, A334302, A334433, A334438, A334439, A334440.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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