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