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%I #18 Oct 10 2023 09:30:12
%S 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,
%T 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,
%U 11,10,10,10,10,10,10,10,10,11,10,12,13,11,13,11,12,15,12,11,13,11,13,12
%N Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.
%C Warning: Do not confuse with the non-strict version A046663.
%C Rows are palindromes.
%H P. Erdős, J. L. Nicolas and A. Sárközy, <a href="http://dx.doi.org/10.1016/0012-365X(89)90086-1">On the number of partitions of n without a given subsum (I)</a>, Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.
%e Triangle begins:
%e 1
%e 1 1
%e 1 2 1
%e 2 2 2 2
%e 2 2 3 2 2
%e 3 3 3 3 3 3
%e 3 4 3 5 3 4 3
%e 5 5 4 5 5 4 5 5
%e 5 6 5 6 7 6 5 6 5
%e 7 7 7 7 7 7 7 7 7 7
%e 8 9 8 8 8 11 8 8 8 9 8
%e Row n = 8 counts the following strict partitions:
%e (8) (8) (8) (8) (8) (8) (8)
%e (6,2) (7,1) (7,1) (7,1) (7,1) (7,1) (6,2)
%e (5,3) (5,3) (6,2) (6,2) (6,2) (5,3) (5,3)
%e (4,3,1) (5,3) (4,3,1)
%e (5,2,1)
%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]
%Y Columns k = 0 and k = n are A025147.
%Y The non-strict version is A046663, central column A006827.
%Y Central column n = 2k is A321142.
%Y The complement for subsets instead of strict partitions is A365381.
%Y The complement is A365661, non-strict A365543, central column A237258.
%Y Row sums are A365922.
%Y A000009 counts subsets summing to n.
%Y A000124 counts distinct possible sums of subsets of {1..n}.
%Y A124506 appears to count combination-free subsets, differences of A326083.
%Y A364272 counts sum-full strict partitions, sum-free A364349.
%Y A364350 counts combination-free strict partitions, complement A364839.
%Y Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.
%K nonn,tabl
%O 2,5
%A _Gus Wiseman_, Sep 17 2023
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