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A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

%I A331581 #7 May 09 2020 07:24:18
%S A331581 0,1,2,1,3,2,1,4,3,2,2,1,5,4,3,3,2,2,1,6,5,4,4,3,3,3,2,2,2,1,7,6,5,5,
%T A331581 4,4,4,3,3,3,3,2,2,2,1,8,7,6,6,5,5,5,4,4,4,4,4,3,3,3,3,3,2,2,2,2,1,9,
%U A331581 8,7,7,6,6,6,5,5,5,5,5,4,4,4,4,4,4,3,3,3,3,3,3,3,2,2,2,2,1
%N A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.
%C A331581 The first partition ranked by A080577 is (); there is no zeroth partition.
%H A331581 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A331581 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A331581 a(n) = A061395(A129129(n - 1)).
%e A331581 The sequence of all partitions in graded reverse-lexicographic order begins as follows. The terms are the initial parts.
%e A331581   ()         (3,2)        (2,1,1,1,1)    (2,2,1,1,1)
%e A331581   (1)        (3,1,1)      (1,1,1,1,1,1)  (2,1,1,1,1,1)
%e A331581   (2)        (2,2,1)      (7)            (1,1,1,1,1,1,1)
%e A331581   (1,1)      (2,1,1,1)    (6,1)          (8)
%e A331581   (3)        (1,1,1,1,1)  (5,2)          (7,1)
%e A331581   (2,1)      (6)          (5,1,1)        (6,2)
%e A331581   (1,1,1)    (5,1)        (4,3)          (6,1,1)
%e A331581   (4)        (4,2)        (4,2,1)        (5,3)
%e A331581   (3,1)      (4,1,1)      (4,1,1,1)      (5,2,1)
%e A331581   (2,2)      (3,3)        (3,3,1)        (5,1,1,1)
%e A331581   (2,1,1)    (3,2,1)      (3,2,2)        (4,4)
%e A331581   (1,1,1,1)  (3,1,1,1)    (3,2,1,1)      (4,3,1)
%e A331581   (5)        (2,2,2)      (3,1,1,1,1)    (4,2,2)
%e A331581   (4,1)      (2,2,1,1)    (2,2,2,1)      (4,2,1,1)
%e A331581 Triangle begins:
%e A331581   0
%e A331581   1
%e A331581   2 1
%e A331581   3 2 1
%e A331581   4 3 2 2 1
%e A331581   5 4 3 3 2 2 1
%e A331581   6 5 4 4 3 3 3 2 2 2 1
%e A331581   7 6 5 5 4 4 4 3 3 3 3 2 2 2 1
%e A331581   8 7 6 6 5 5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 2 1
%t A331581 revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
%t A331581 Prepend[First/@Join@@Table[Sort[IntegerPartitions[n],revlexsort],{n,8}],0]
%Y A331581 Row lengths are A000041.
%Y A331581 Lexicographically ordered reversed partitions are A026791.
%Y A331581 Reverse-colexicographically ordered partitions are A026792.
%Y A331581 Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
%Y A331581 The version for compositions is A065120 or A333766.
%Y A331581 Reverse-lexicographically ordered partitions are A080577.
%Y A331581 Distinct parts of these partitions are counted by A115623.
%Y A331581 Lexicographically ordered partitions are A193073.
%Y A331581 Colexicographically ordered partitions are A211992.
%Y A331581 Lengths of these partitions are A238966.
%Y A331581 Cf. A036037, A048793, A063008, A066099, A129129, A185974, A228100, A228531, A334301, A334434, A334436, A334438.
%K A331581 nonn,tabf
%O A331581 1,3
%A A331581 _Gus Wiseman_, May 08 2020