A334442 - OEIS

%I #20 Sep 22 2023 08:46:46

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

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

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

%N Irregular triangle whose reversed rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

%C First differs from A036036 for reversed partitions of 9. Namely, this sequence has (2,2,5) before (1,4,4), while A036036 has (1,4,4) before (2,2,5).

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

%e The sequence of all partitions begins:

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

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

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

%e   (1,1)      (1,1,1,2)    (1,6)          (8)

%e   (3)        (1,1,1,1,1)  (2,5)          (1,7)

%e   (1,2)      (6)          (3,4)          (2,6)

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

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

%e   (1,3)      (3,3)        (1,3,3)        (1,1,6)

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

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

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

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

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

%e This sequence can also be interpreted as the following triangle:

%e                   0

%e                  (1)

%e                (2)(11)

%e              (3)(12)(111)

%e         (4)(13)(22)(112)(1111)

%e   (5)(14)(23)(113)(122)(1112)(11111)

%e Taking Heinz numbers (A334438) gives:

%e    1

%e    2

%e    3   4

%e    5   6   8

%e    7  10   9  12  16

%e   11  14  15  20  18  24  32

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

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

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

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

%o (PARI) A334442_row(n)=vecsort(partitions(n),p->concat(#p,-Vecrev(p))) \\ Rows of triangle defined in EXAMPLE (all partitions of n). Wrap into [Vec(p)|p<-...] to avoid "Vecsmall". - _M. F. Hasler_, May 14 2020

%Y Row lengths are A036043.

%Y The version for reversed partitions is A334301.

%Y The version for colex instead of revlex is A334302.

%Y Taking Heinz numbers gives A334438.

%Y The version with rows reversed is A334439.

%Y Ignoring length gives A335122.

%Y Lexicographically ordered reversed partitions are A026791.

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

%Y Partitions in increasing-length colex order (sum/length/colex) are A036037.

%Y Reverse-lexicographically ordered partitions are A080577.

%Y Lexicographically ordered partitions are A193073.

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

%Y Sorting partitions by Heinz number gives A296150.

%Y Cf. A026791, A112798, A124734, A129129, A185974, A228100, A228531, A296774, A334433, A334435, A334436.

%K nonn,tabf

%O 0,2

%A _Gus Wiseman_, May 07 2020