A333486 - OEIS

%I #11 Sep 22 2023 08:47:51

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

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

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

%N Length of the n-th reversed integer partition in graded reverse-lexicographic order. Partition lengths of A228531.

%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 Triangle begins:

%e   0

%e   1

%e   1 2

%e   1 2 3

%e   1 2 2 3 4

%e   1 2 2 3 3 4 5

%e   1 2 2 3 2 3 3 4 4 5 6

%e   1 2 2 3 2 3 3 4 3 4 4 5 5 6 7

%e   1 2 2 2 3 3 4 2 3 3 4 3 4 4 5 4 5 5 6 6 7 8

%t revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];

%t Table[Length/@Sort[Reverse/@IntegerPartitions[n],revlexsort],{n,0,8}]

%Y Row lengths are A000041.

%Y The generalization to compositions is A000120.

%Y Row sums are A006128.

%Y The same partition has sum A036042.

%Y The length-sensitive version (sum/length/revlex) is A036043.

%Y The colexicographic version (sum/colex) is A049085.

%Y The same partition has minimum A182715.

%Y The lexicographic version (sum/lex) is A193173.

%Y The tetrangle of these partitions is A228531.

%Y The version for non-reversed partitions is A238966.

%Y The same partition has Heinz number A334436.

%Y Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.

%Y Partitions in lexicographic order (sum/lex) are A193073.

%Y Partitions in colexicographic order (sum/colex) are A211992.

%Y Partitions in opposite Abramowitz-Stegun order (sum/length/revlex) are A334439.

%Y Cf. A026792, A049085, A080576, A080577, A103921, A112798, A115623, A129129, A331581, A334302, A334435, A334442.

%K nonn,tabf

%O 0,4

%A _Gus Wiseman_, May 23 2020