cp's OEIS Frontend

A330371 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by the lower value of their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = n..1.

%I A330371 #28 Aug 17 2020 23:29:28
%S A330371 1,2,1,1,3,2,1,1,1,1,4,3,1,2,2,2,1,1,1,1,1,1,5,4,1,3,2,3,1,1,2,2,1,2,
%T A330371 1,1,1,1,1,1,1,1,6,5,1,4,2,4,1,1,3,3,3,2,1,2,2,2,3,1,1,1,2,2,1,1,2,1,
%U A330371 1,1,1,1,1,1,1,1,1,7,6,1,5,2,5,1,1,4,3,4,2,1,3,3,1,4,1,1,1,3,2,2,3,2,1,1
%N A330371 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by the lower value of their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = n..1.
%C A330371 In this triangle the partitions of n are ordered by their n-th rank. The partitions that have the same n-th rank appears ordered by their (n-1)-st rank. The partitions that have the same n-th rank and the same (n-1)-st rank appears ordered by their (n-2)-nd rank, and so on. The partitions that have all k-ranks equal zero appears ordered by their largest parts, then by their second largest parts, then by their third largest parts, and so on.
%C A330371 Note that a partition and its conjugate partition both are equidistants from the center of the list of partitions of n.
%C A330371 For further information see A330370.
%C A330371 First differs from A036037, A181317, A330370 and A334439 at a(48).
%C A330371 First differs from A080577 at a(56).
%e A330371 Triangle begins:
%e A330371 [1];
%e A330371 [2], [1,1];
%e A330371 [3], [2,1], [1,1,1];
%e A330371 [4], [3,1], [2,2], [2,1,1], [1,1,1,1];
%e A330371 [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
%e A330371 [6], [5,1], [4,2], [4,1,1], [3,3], [3,2,1], [2,2,2], [3,1,1,1], [2,2,1,1], ...
%e A330371 .
%e A330371 For n = 9 the 9th row of the triangle contains the partitions ordered as shown below:
%e A330371 ---------------------------------------------------------------------------------
%e A330371                                                                 Ranks
%e A330371           Conjugate
%e A330371 Label     with label    Partition                 k = 1  2  3  4  5  6  7  8  9
%e A330371 ---------------------------------------------------------------------------------
%e A330371    1         30         [9]                           8 -1 -1 -1 -1 -1 -1 -1 -1
%e A330371    2         29         [8, 1]                        6  0 -1 -1 -1 -1 -1 -1  0
%e A330371    3         28         [7, 2]                        5  0 -1 -1 -1 -1 -1  0  0
%e A330371    4         27         [7, 1, 1]                     4  0  0 -1 -1 -1 -1  0  0
%e A330371    5         26         [6, 3]                        4  1 -2 -1 -1 -1  0  0  0
%e A330371    6         25         [6, 2, 1]                     3  0  0 -1 -1 -1  0  0  0
%e A330371    7         24         [6, 1, 1, 1]                  2  0  0  0 -1 -1  0  0  0
%e A330371    8         23         [5, 4]                        3  2 -2 -2 -1  0  0  0  0
%e A330371    9         22         [5, 3, 1]                     2  1 -1 -1 -1  0  0  0  0
%e A330371   10         21         [5, 2, 2]                     2 -1  1 -1 -1  0  0  0  0
%e A330371   11         20         [5, 2, 1, 1]                  1  0  0  0 -1  0  0  0  0
%e A330371   12         19         [4, 4, 1]                     1  2 -1 -2  0  0  0  0  0
%e A330371   13         18         [4, 3, 2]                     1  0  0 -1  0  0  0  0  0
%e A330371   14         17         [4, 3, 1, 1]                  0  1 -1  0  0  0  0  0  0
%e A330371   15  (self-conjugate)  [5, 1, 1, 1, 1]  All zeros -> 0  0  0  0  0  0  0  0  0
%e A330371   16  (self-conjugate)  [3, 3, 3]        All zeros -> 0  0  0  0  0  0  0  0  0
%e A330371   17         14         [4, 2, 2, 1]                  0 -1  1  0  0  0  0  0  0
%e A330371   18         13         [3, 3, 2, 1]                 -1  0  0  1  0  0  0  0  0
%e A330371   19         12         [3, 2, 2, 2]                 -1 -2  1  2  0  0  0  0  0
%e A330371   20         11         [4, 2, 1, 1, 1]              -1  0  0  0  1  0  0  0  0
%e A330371   21         10         [3, 3, 1, 1, 1]              -2  1 -1  1  1  0  0  0  0
%e A330371   22          9         [3, 2, 2, 1, 1]              -2 -1  1  1  1  0  0  0  0
%e A330371   23          8         [2, 2, 2, 2, 1]              -3 -2  2  2  1  0  0  0  0
%e A330371   24          7         [4, 1, 1, 1, 1, 1]           -2  0  0  0  1  1  0  0  0
%e A330371   25          6         [3, 2, 1, 1, 1, 1]           -3  0  0  1  1  1  0  0  0
%e A330371   26          5         [2, 2, 2, 1, 1, 1]           -4 -1  2  1  1  1  0  0  0
%e A330371   27          4         [3, 1, 1, 1, 1, 1, 1]        -4  0  0  1  1  1  1  0  0
%e A330371   28          3         [2, 2, 1, 1, 1, 1, 1]        -5  0  1  1  1  1  1  0  0
%e A330371   29          2         [2, 1, 1, 1, 1, 1, 1, 1]     -6  0  1  1  1  1  1  1  0
%e A330371   30          1         [1, 1, 1, 1, 1, 1, 1, 1, 1]  -8  1  1  1  1  1  1  1  1
%Y A330371 Another version of A330370.
%Y A330371 Row n contains A000041(n) partitions.
%Y A330371 Row n has length A006128(n).
%Y A330371 The sum of n-th row is A066186(n).
%Y A330371 Cf. A000700, A036037, A080577, A181317, A207031, A330368, A330370, A334439.
%Y A330371 For the "k-th rank" see also: A181187, A208478, A208479, A208482, A208483.
%K A330371 nonn,tabf
%O A330371 1,2
%A A330371 _Omar E. Pol_, Dec 15 2019