A334302 - OEIS

%I #12 May 28 2020 05:03:05

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

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

%U 1,1,2,1,1,1,1,1,1,7,3,4,2,5,1,6,2,2,3

%N Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

%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 The sequence of all reversed partitions begins:

%e   ()         (1,4)        (1,1,1,1,2)

%e   (1)        (1,2,2)      (1,1,1,1,1,1)

%e   (2)        (1,1,3)      (7)

%e   (1,1)      (1,1,1,2)    (3,4)

%e   (3)        (1,1,1,1,1)  (2,5)

%e   (1,2)      (6)          (1,6)

%e   (1,1,1)    (3,3)        (2,2,3)

%e   (4)        (2,4)        (1,3,3)

%e   (2,2)      (1,5)        (1,2,4)

%e   (1,3)      (2,2,2)      (1,1,5)

%e   (1,1,2)    (1,2,3)      (1,2,2,2)

%e   (1,1,1,1)  (1,1,4)      (1,1,2,3)

%e   (5)        (1,1,2,2)    (1,1,1,4)

%e   (2,3)      (1,1,1,3)    (1,1,1,2,2)

%e This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.

%e                             0

%e                            (1)

%e                         (2) (1,1)

%e                     (3) (1,2) (1,1,1)

%e             (4) (2,2) (1,3) (1,1,2) (1,1,1,1)

%e   (5) (2,3) (1,4) (1,2,2) (1,1,3) (1,1,1,2) (1,1,1,1,1)

%e Showing partitions as their Heinz numbers (see A334435) gives:

%e    1

%e    2

%e    3   4

%e    5   6   8

%e    7   9  10  12  16

%e   11  15  14  18  20  24  32

%e   13  25  21  22  27  30  28  36  40  48  64

%e   17  35  33  26  45  50  42  44  54  60  56  72  80  96 128

%t revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];

%t Join@@Table[Sort[Sort/@IntegerPartitions[n],revlensort],{n,0,8}]

%Y Row lengths are A036043.

%Y Lexicographically ordered reversed partitions are A026791.

%Y The dual ordering (sum/length/lex) of reversed partitions is A036036.

%Y Reverse-lexicographically ordered partitions are A080577.

%Y Sorting reversed partitions by Heinz number gives A112798.

%Y Lexicographically ordered partitions are A193073.

%Y Graded Heinz numbers are A215366.

%Y Ignoring length gives A228531.

%Y Sorting partitions by Heinz number gives A296150.

%Y The version for compositions is A296774.

%Y The dual ordering (sum/length/lex) of non-reversed partitions is A334301.

%Y Taking Heinz numbers gives A334435.

%Y The version for regular (non-reversed) partitions is A334439 (not A036037).

%Y Cf. A000041, A048793, A066099, A080576, A124734, A162247, A211992, A228100, A228351.

%K nonn,tabf

%O 0,2

%A _Gus Wiseman_, Apr 30 2020