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

%I A365663 #23 Apr 05 2025 23:17:57
%S A365663 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 A365663 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 A365663 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 A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.
%C A365663 Warning: Do not confuse with the non-strict version A046663.
%C A365663 Rows are palindromes.
%H A365663 Robert Price, <a href="/A365663/b365663.txt">Table of n, a(n) for n = 2..1226</a>
%H A365663 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 A365663 Triangle begins:
%e A365663   1
%e A365663   1  1
%e A365663   1  2  1
%e A365663   2  2  2  2
%e A365663   2  2  3  2  2
%e A365663   3  3  3  3  3  3
%e A365663   3  4  3  5  3  4  3
%e A365663   5  5  4  5  5  4  5  5
%e A365663   5  6  5  6  7  6  5  6  5
%e A365663   7  7  7  7  7  7  7  7  7  7
%e A365663   8  9  8  8  8 11  8  8  8  9  8
%e A365663 Row n = 8 counts the following strict partitions:
%e A365663   (8)    (8)      (8)    (8)      (8)    (8)      (8)
%e A365663   (6,2)  (7,1)    (7,1)  (7,1)    (7,1)  (7,1)    (6,2)
%e A365663   (5,3)  (5,3)    (6,2)  (6,2)    (6,2)  (5,3)    (5,3)
%e A365663          (4,3,1)         (5,3)           (4,3,1)
%e A365663                          (5,2,1)
%t A365663 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]
%Y A365663 Columns k = 0 and k = n are A025147.
%Y A365663 The non-strict version is A046663, central column A006827.
%Y A365663 Central column n = 2k is A321142.
%Y A365663 The complement for subsets instead of strict partitions is A365381.
%Y A365663 The complement is A365661, non-strict A365543, central column A237258.
%Y A365663 Row sums are A365922.
%Y A365663 A000009 counts subsets summing to n.
%Y A365663 A000124 counts distinct possible sums of subsets of {1..n}.
%Y A365663 A124506 appears to count combination-free subsets, differences of A326083.
%Y A365663 A364272 counts sum-full strict partitions, sum-free A364349.
%Y A365663 A364350 counts combination-free strict partitions, complement A364839.
%Y A365663 Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.
%K A365663 nonn,tabl
%O A365663 2,5
%A A365663 _Gus Wiseman_, Sep 17 2023