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A335122 Irregular triangle whose reversed rows are all integer partitions in graded reverse-lexicographic order.

%I A335122 #13 Sep 22 2023 08:58:26
%S A335122 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 A335122 1,1,2,1,1,1,1,1,6,1,5,2,4,1,1,4,3,3,1,2,3,1,1,1,3,2,2,2,1,1,2,2,1,1,
%U A335122 1,1,2,1,1,1,1,1,1,7,1,6,2,5,1,1,5,3,4,1,2,4
%N A335122 Irregular triangle whose reversed rows are all integer partitions in graded reverse-lexicographic order.
%C A335122 First differs from A036036 for partitions of 6.
%C A335122 First differs from A334442 for partitions of 6.
%C A335122 Also reversed partitions in reverse-colexicographic order.
%H A335122 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A335122 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A335122 The sequence of all reversed partitions begins:
%e A335122   ()         (1,1,3)        (7)              (8)
%e A335122   (1)        (1,2,2)        (1,6)            (1,7)
%e A335122   (2)        (1,1,1,2)      (2,5)            (2,6)
%e A335122   (1,1)      (1,1,1,1,1)    (1,1,5)          (1,1,6)
%e A335122   (3)        (6)            (3,4)            (3,5)
%e A335122   (1,2)      (1,5)          (1,2,4)          (1,2,5)
%e A335122   (1,1,1)    (2,4)          (1,1,1,4)        (1,1,1,5)
%e A335122   (4)        (1,1,4)        (1,3,3)          (4,4)
%e A335122   (1,3)      (3,3)          (2,2,3)          (1,3,4)
%e A335122   (2,2)      (1,2,3)        (1,1,2,3)        (2,2,4)
%e A335122   (1,1,2)    (1,1,1,3)      (1,1,1,1,3)      (1,1,2,4)
%e A335122   (1,1,1,1)  (2,2,2)        (1,2,2,2)        (1,1,1,1,4)
%e A335122   (5)        (1,1,2,2)      (1,1,1,2,2)      (2,3,3)
%e A335122   (1,4)      (1,1,1,1,2)    (1,1,1,1,1,2)    (1,1,3,3)
%e A335122   (2,3)      (1,1,1,1,1,1)  (1,1,1,1,1,1,1)  (1,2,2,3)
%e A335122 We have the following tetrangle of reversed partitions:
%e A335122                              0
%e A335122                             (1)
%e A335122                           (2)(11)
%e A335122                         (3)(12)(111)
%e A335122                    (4)(13)(22)(112)(1111)
%e A335122              (5)(14)(23)(113)(122)(1112)(11111)
%e A335122   (6)(15)(24)(114)(33)(123)(1113)(222)(1122)(11112)(111111)
%t A335122 revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
%t A335122 Reverse/@Join@@Table[Sort[IntegerPartitions[n],revlexsort],{n,0,8}]
%Y A335122 Row lengths are A000041.
%Y A335122 The version for reversed partitions is A026792.
%Y A335122 The version for colex instead of revlex is A026791.
%Y A335122 The version for lex instead of revlex is A080576.
%Y A335122 The non-reflected version is A080577.
%Y A335122 The number of distinct parts is A115623.
%Y A335122 Taking Heinz numbers gives A129129.
%Y A335122 The version for compositions is A228351.
%Y A335122 Partition lengths are A238966.
%Y A335122 Partition maxima are A331581.
%Y A335122 The length-sensitive version is A334442.
%Y A335122 Lexicographically ordered partitions are A193073.
%Y A335122 Partitions in colexicographic order are A211992.
%Y A335122 Cf. A036036, A036037, A112798, A129129, A228531, A296774, A334301, A334302, A334435, A334436, A334438, A334439.
%K A335122 nonn,tabf
%O A335122 0,2
%A A335122 _Gus Wiseman_, May 24 2020