A355147 Triangle read by rows: T(n,k) is the number of product-free subsets of {1,...,n} with cardinality k; n >= 0, 0 <= k <= A028391(n).

1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 5, 2, 1, 5, 9, 6, 1, 1, 6, 14, 15, 7, 1, 1, 7, 20, 29, 22, 8, 1, 1, 8, 26, 43, 38, 17, 3, 1, 9, 34, 68, 76, 47, 15, 2, 1, 10, 43, 102, 144, 123, 62, 17, 2, 1, 11, 53, 143, 234, 238, 149, 55, 11, 1, 1, 12, 64, 196, 377, 472, 387, 204, 66, 12, 1

Offset: 0

Author

Marcel K. Goh, Jun 28 2022

Keywords

• nonn
• tabf

Comments

S is product-free if for any i,j in S, not necessarily distinct, i*j is not in S.

For n >= 2, the alternating row sums give 0.

Examples

			Triangle T(n,k) begins:
  n/k 0  1  2   3   4   5   6  7  8 9
   0  1
   1  1
   2  1  1
   3  1  2  1
   4  1  3  2
   5  1  4  5   2
   6  1  5  9   6   1
   7  1  6 14  15   7   1
   8  1  7 20  29  22   8   1
   9  1  8 26  43  38  17   3
  10  1  9 34  68  76  47  15  2
  11  1 10 43 102 144 123  62 17  2
  12  1 11 53 143 234 238 149 55 11 1
  ...
For n=5 and k=3 the T(5,3) = 2 sets are {2,3,5} and {3,4,5}.
		

Links

• Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.

Crossrefs

Row sums give A326489.

Cf. A028391.