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A334302 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. 35

%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

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