cp's OEIS Frontend

A367405 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.

%I A367405 #6 Nov 19 2023 10:34:28
%S A367405 1,0,1,0,0,2,1,1,1,2,1,0,1,1,3,1,1,1,1,2,3,1,1,1,2,2,2,4,2,2,3,2,3,2,
%T A367405 3,4,2,2,3,2,3,3,3,3,5,3,2,4,3,4,4,5,3,4,5,3,3,5,4,4,5,5,5,4,4,6,4,3,
%U A367405 6,5,6,5,7,5,7,4,5,6,5,5,7,7,8,7,8,8,7,7,5,5,7
%N A367405 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.
%e A367405 Triangle begins:
%e A367405   1
%e A367405   0  1
%e A367405   0  0  2
%e A367405   1  1  1  2
%e A367405   1  0  1  1  3
%e A367405   1  1  1  1  2  3
%e A367405   1  1  1  2  2  2  4
%e A367405   2  2  3  2  3  2  3  4
%e A367405   2  2  3  2  3  3  3  3  5
%e A367405   3  2  4  3  4  4  5  3  4  5
%e A367405   3  3  5  4  4  5  5  5  4  4  6
%e A367405   4  3  6  5  6  5  7  5  7  4  5  6
%e A367405   5  5  7  7  8  7  8  8  7  7  5  5  7
%e A367405   6  5  9  8 10  7 10  9 10  7  9  5  6  7
%e A367405   7  7 10 10 12 11 11 11 12 10  9  9  6  6  8
%e A367405   9  7 13 11 15 12 13 13 15 13 13  9 11  6  7  8
%e A367405 Row n = 9 counts the following strict partitions:
%e A367405   (6,2,1)  (5,3,1)  (4,3,2)  (5,3,1)  (6,2,1)  (6,2,1)  (8,1)
%e A367405                              (4,3,2)  (4,3,2)  (5,3,1)  (7,2)
%e A367405                                                         (6,3)
%e A367405                                                         (5,4)
%e A367405 Row n = 13 counts the following strict partitions (A=10, B=11, C=12):
%e A367405   A21   931   841   751   652   751   841   931   A21  A21  C1
%e A367405   7321  7321  832   742   643   7321  742   832   832  931  B2
%e A367405   6421  5431  7321  6421  6421  652   7321  7321  742  841  A3
%e A367405               6421  5431  5431  6421  643   643   652  751  94
%e A367405               5431              5431  5431  6421            85
%e A367405                                                             76
%t A367405 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#,{2}], k]&]], {n,3,10}, {k,3,n}]
%Y A367405 Column n = k is A004526.
%Y A367405 Column k = 3 is A025148.
%Y A367405 For subsets instead of partitions we have A365541, non-binary A365381.
%Y A367405 The non-binary version is A365661, non-strict A365543.
%Y A367405 The non-binary complement is A365663, non-strict A046663.
%Y A367405 Row sums are A366741, non-strict A366738.
%Y A367405 The non-strict version is A367404.
%Y A367405 Cf. A000041, A088809, A093971, A122768, A108917, A284640, A304792, A364272, A364911, A365658.
%K A367405 nonn,tabl
%O A367405 3,6
%A A367405 _Gus Wiseman_, Nov 18 2023