cp's OEIS Frontend

A330372 Irregular triangle read by rows in which row n lists the self-conjugate partitions of n, ordered by their k-th largest parts, or 0 if such partitions does not exist.

%I A330372 #50 Feb 20 2020 13:57:32
%S A330372 0,1,0,2,1,2,2,3,1,1,3,2,1,4,1,1,1,4,2,1,1,3,3,2,5,1,1,1,1,3,3,3,5,2,
%T A330372 1,1,1,4,3,2,1,6,1,1,1,1,1,4,3,3,1,6,2,1,1,1,1,5,3,2,1,1,4,4,2,2,7,1,
%U A330372 1,1,1,1,1,5,3,3,1,1,4,4,3,2
%N A330372 Irregular triangle read by rows in which row n lists the self-conjugate partitions of n, ordered by their k-th largest parts, or 0 if such partitions does not exist.
%C A330372 Row n lists the partitions of n whose Ferrers diagrams are symmetrics.
%C A330372 The k-th part of a partition equals the number of parts >= k of its conjugate partition. Hence, the k-th part of a self-conjugate partition equals the number of parts >= k.
%C A330372 The k-th rank of a partition is the k-th part minus the number of parts >= k. Thus all ranks of a conjugate-partitions are zero. Therefore row n lists the partitions of n whose n ranks are zero, n >= 1. For more information about the k-th ranks see A208478.
%H A330372 Freddy Barrera, <a href="/A330372/b330372.txt">Rows n = 0..50, flattened</a>
%e A330372 Triangle begins (rows n = 0..10):
%e A330372 [0];
%e A330372 [1];
%e A330372 [0];
%e A330372 [2, 1];
%e A330372 [2, 2];
%e A330372 [3, 1, 1];
%e A330372 [3, 2, 1];
%e A330372 [4, 1, 1, 1];
%e A330372 [4, 2, 1, 1], [3, 3, 2];
%e A330372 [5, 1, 1, 1, 1], [3, 3, 3];
%e A330372 [5, 2, 1, 1, 1], [4, 3, 2, 1];
%e A330372 ...
%e A330372 For n = 10 there are only two partitions of 10 whose Ferrers diagram are symmetric, they are [5, 2, 1, 1, 1] and [4, 3, 2, 1] as shown below:
%e A330372   * * * * *
%e A330372   * *
%e A330372   *
%e A330372   *
%e A330372   *
%e A330372             * * * *
%e A330372             * * *
%e A330372             * *
%e A330372             *
%e A330372 So these partitions form the 10th row of triangle.
%e A330372 On the other hand, only two partitions of 10 have all their ranks equal to zero, they are [5, 2, 1, 1, 1] and [4, 3, 2, 1], so these partitions form the 10th row of triangle.
%Y A330372 Row n contains A000700(n) partitions.
%Y A330372 The number of positive terms in row n is A067619(n).
%Y A330372 Row sums give A330373.
%Y A330372 Column 2 gives A000034.
%Y A330372 Column 3 gives A000012.
%Y A330372 For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483, A330370.
%K A330372 nonn,tabf
%O A330372 0,4
%A A330372 _Omar E. Pol_, Dec 17 2019
%E A330372 More terms from _Freddy Barrera_, Dec 31 2019