A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12

Offset: 2

Author

Gus Wiseman, Sep 17 2023

Keywords

• nonn
• tabl

Comments

Warning: Do not confuse with the non-strict version A046663.

Rows are palindromes.

Examples

			Triangle begins:
  1
  1  1
  1  2  1
  2  2  2  2
  2  2  3  2  2
  3  3  3  3  3  3
  3  4  3  5  3  4  3
  5  5  4  5  5  4  5  5
  5  6  5  6  7  6  5  6  5
  7  7  7  7  7  7  7  7  7  7
  8  9  8  8  8 11  8  8  8  9  8
Row n = 8 counts the following strict partitions:
  (8)    (8)      (8)    (8)      (8)    (8)      (8)
  (6,2)  (7,1)    (7,1)  (7,1)    (7,1)  (7,1)    (6,2)
  (5,3)  (5,3)    (6,2)  (6,2)    (6,2)  (5,3)    (5,3)
         (4,3,1)         (5,3)           (4,3,1)
                         (5,2,1)
		

Links

• Robert Price, Table of n, a(n) for n = 2..1226
• P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Crossrefs

Columns k = 0 and k = n are A025147.

The non-strict version is A046663, central column A006827.

Central column n = 2k is A321142.

The complement for subsets instead of strict partitions is A365381.

The complement is A365661, non-strict A365543, central column A237258.

Row sums are A365922.

A000009 counts subsets summing to n.

A000124 counts distinct possible sums of subsets of {1..n}.

A124506 appears to count combination-free subsets, differences of A326083.

A364272 counts sum-full strict partitions, sum-free A364349.

A364350 counts combination-free strict partitions, complement A364839.

Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]