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A334301 Irregular triangle read by rows where row k is the k-th integer partition, if partitions are sorted first by sum, then by length, and finally lexicographically. 41

%I #13 May 31 2020 07:09:36

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

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

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

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

%C This is the Abramowitz-Stegun ordering of integer partitions when they are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A036036.

%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>

%e The sequence of all partitions in Abramowitz-Stegun order begins:

%e () (41) (21111) (31111) (3221)

%e (1) (221) (111111) (211111) (3311)

%e (2) (311) (7) (1111111) (4211)

%e (11) (2111) (43) (8) (5111)

%e (3) (11111) (52) (44) (22211)

%e (21) (6) (61) (53) (32111)

%e (111) (33) (322) (62) (41111)

%e (4) (42) (331) (71) (221111)

%e (22) (51) (421) (332) (311111)

%e (31) (222) (511) (422) (2111111)

%e (211) (321) (2221) (431) (11111111)

%e (1111) (411) (3211) (521) (9)

%e (5) (2211) (4111) (611) (54)

%e (32) (3111) (22111) (2222) (63)

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

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

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

%e Showing partitions as their Heinz numbers (see A334433) 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 Join@@Table[Sort[IntegerPartitions[n]],{n,0,8}]

%Y Lexicographically ordered reversed partitions are A026791.

%Y The version for reversed partitions (sum/length/lex) is A036036.

%Y Row lengths are A036043.

%Y Reverse-lexicographically ordered partitions are A080577.

%Y The version for compositions is A124734.

%Y Lexicographically ordered partitions are A193073.

%Y Sorting by Heinz number gives A296150, or A112798 for reversed partitions.

%Y Sorting first by sum, then by Heinz number gives A215366.

%Y Reversed partitions under the dual ordering (sum/length/revlex) are A334302.

%Y Taking Heinz numbers gives A334433.

%Y The reverse-lexicographic version is A334439 (not A036037).

%Y Cf. A000041, A048793, A066099, A162247, A211992, A228100, A228351, A228531.

%K nonn,tabf

%O 0,2

%A _Gus Wiseman_, Apr 29 2020

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