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A330369 Triangle read by rows: T(n,k) (1 <= k <= n) is the total number of right angles of size k in all partitions of n.

%I A330369 #61 Jul 09 2025 04:50:11
%S A330369 1,0,2,0,0,3,1,0,1,4,2,0,0,2,5,3,2,0,2,3,6,4,4,0,0,4,4,7,5,6,3,0,3,6,
%T A330369 5,8,7,8,7,0,1,6,8,6,9,9,10,11,4,0,6,9,10,7,10,13,12,15,10,0,2,11,12,
%U A330369 12,8,11
%N A330369 Triangle read by rows: T(n,k) (1 <= k <= n) is the total number of right angles of size k in all partitions of n.
%C A330369 This triangle has the property that it contains the triangle A049597, since if we replace with zeros the positive terms before the first zero in the row n of this triangle, we get the triangle A049597.
%C A330369 Hence the sum of the terms after the last zero in row n equals A000041(n), the number of partitions of n (see the Example section).
%C A330369 Observation: at least the first 11 terms of column 1 coincide with A188674 (using the same indices).
%D A330369 G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143 [Defines the right angles in the Ferrers graph of a partition. - _N. J. A. Sloane_, Nov 20 2020]
%e A330369 Triangle begins:
%e A330369    1;
%e A330369    0,  2;
%e A330369    0,  0,  3;
%e A330369    1,  0,  1,  4;
%e A330369    2,  0,  0,  2,  5;
%e A330369    3,  2,  0,  2,  3,  6;
%e A330369    4,  4,  0,  0,  4,  4,  7;
%e A330369    5,  6,  3,  0,  3,  6,  5,  8;
%e A330369    7,  8,  7,  0,  1,  6,  8,  6,  9;
%e A330369    9, 10, 11,  4,  0,  6,  9, 10,  7, 10;
%e A330369   13, 12, 15, 10,  0,  2, 11, 12, 12,  8, 11;
%e A330369 Figure 1 below shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. Figure 2 shows the right-angles diagram of the same partition. Note that in this last diagram we can see the size of the three right angles as follows: the first right angle has size 14 because it contains 14 square cells, the second right angle has size 8 and the third right angle has size 2.
%e A330369 .
%e A330369 .                                     Right-angles   Right
%e A330369 Part   Ferrers diagram         Part   diagram        angle
%e A330369                                       _ _ _ _ _ _ _
%e A330369   7    * * * * * * *             7   |  _ _ _ _ _ _|  14
%e A330369   6    * * * * * *               6   | |  _ _ _ _|     8
%e A330369   3    * * *                     3   | | | |           2
%e A330369   3    * * *                     3   | | |_|
%e A330369   2    * *                       2   | |_|
%e A330369   1    *                         1   | |
%e A330369   1    *                         1   | |
%e A330369   1    *                         1   |_|
%e A330369 .
%e A330369        Figure 1.                      Figure 2.
%e A330369 .
%e A330369 For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows:
%e A330369 .
%e A330369     _       _ _       _ _ _       _ _ _ _       _ _ _ _ _
%e A330369   1| |8   2|  _|8   3|  _ _|8   4|  _ _ _|8   5|  _ _ _ _|8
%e A330369   1| |    1| |      1| |        1| |          1| |
%e A330369   1| |    1| |      1| |        1| |          1| |
%e A330369   1| |    1| |      1| |        1| |          1|_|
%e A330369   1| |    1| |      1| |        1|_|
%e A330369   1| |    1| |      1|_|
%e A330369   1| |    1|_|
%e A330369   1|_|
%e A330369     _ _ _ _ _ _       _ _ _ _ _ _ _       _ _ _ _ _ _ _ _
%e A330369   6|  _ _ _ _ _|8   7|  _ _ _ _ _ _|8   8|_ _ _ _ _ _ _ _|8
%e A330369   1| |              1|_|
%e A330369   1|_|
%e A330369 .
%e A330369     _ _       _ _ _       _ _ _ _       _ _ _ _ _       _ _ _ _ _ _
%e A330369   2|  _|7   3|  _ _|7   4|  _ _ _|7   5|  _ _ _ _|7   6|  _ _ _ _ _|7
%e A330369   2| |_|1   2| |_|  1   2| |_|    1   2| |_|      1   2|_|_|        1
%e A330369   1| |      1| |        1| |          1|_|
%e A330369   1| |      1| |        1|_|
%e A330369   1| |      1|_|
%e A330369   1|_|
%e A330369 .
%e A330369     _ _       _ _ _       _ _ _       _ _ _ _       _ _ _ _       _ _ _ _ _
%e A330369   2|  _|6   3|  _ _|6   3|  _ _|6   4|  _ _ _|6   4|  _ _ _|6   5|  _ _ _ _|6
%e A330369   2| | |2   2| | |  2   3| |_ _|2   2| | |    2   3| |_ _|  2   3|_|_ _|    2
%e A330369   2| |_|    2| |_|      1| |        2|_|_|        1|_|
%e A330369   1| |      1|_|        1|_|
%e A330369   1|_|
%e A330369 .
%e A330369     _ _       _ _ _        _ _ _ _
%e A330369   2|  _|5   3|  _ _|5    4|  _ _ _|5
%e A330369   2| | |3   3| |  _|3    4|_|_ _ _|3
%e A330369   2| | |    2|_|_|
%e A330369   2|_|_|
%e A330369 .
%e A330369 There are  5 right angles of size 1, so T(8,1) = 5.
%e A330369 There are  6 right angles of size 2, so T(8,2) = 6.
%e A330369 There are  3 right angles of size 3, so T(8,3) = 3.
%e A330369 There are no right angle  of size 4, so T(8,4) = 0.
%e A330369 There are  3 right angles of size 5, so T(8,5) = 3.
%e A330369 There are  6 right angles of size 6, so T(8,6) = 6.
%e A330369 There are  5 right angles of size 7, so T(8,7) = 5.
%e A330369 There are  8 right angles of size 8, so T(8,8) = 8.
%e A330369 Hence the 8th row of triangle is [5, 6, 3, 0, 3, 6, 5, 8].
%e A330369 Note that the sum of the terms after the last zero is 3 + 6 + 5 + 8 = 22, equaling A000041(8) = 22, the number of partitions of 8.
%Y A330369 Row sums give A115995.
%Y A330369 Right border gives A000027.
%Y A330369 Cf. A000041, A049597, A083480, A083906, A188674, A259481, A325458, A330375, A330378, A330379.
%K A330369 nonn,tabl,more
%O A330369 1,3
%A A330369 _Omar E. Pol_, Dec 12 2019