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A365543 Triangle read by rows where T(n,k) is the number of integer partitions of n with a submultiset summing to k.

%I A365543 #14 Apr 05 2025 23:17:53
%S A365543 1,1,1,2,1,2,3,2,2,3,5,3,3,3,5,7,5,5,5,5,7,11,7,8,6,8,7,11,15,11,11,
%T A365543 11,11,11,11,15,22,15,17,15,14,15,17,15,22,30,22,23,23,22,22,23,23,22,
%U A365543 30,42,30,33,30,33,25,33,30,33,30,42
%N A365543 Triangle read by rows where T(n,k) is the number of integer partitions of n with a submultiset summing to k.
%C A365543 Rows are palindromic.
%H A365543 Robert Price, <a href="/A365543/b365543.txt">Table of n, a(n) for n = 0..324</a>
%e A365543 Triangle begins:
%e A365543    1
%e A365543    1   1
%e A365543    2   1   2
%e A365543    3   2   2   3
%e A365543    5   3   3   3   5
%e A365543    7   5   5   5   5   7
%e A365543   11   7   8   6   8   7  11
%e A365543   15  11  11  11  11  11  11  15
%e A365543   22  15  17  15  14  15  17  15  22
%e A365543   30  22  23  23  22  22  23  23  22  30
%e A365543   42  30  33  30  33  25  33  30  33  30  42
%e A365543   56  42  45  44  44  43  43  44  44  45  42  56
%e A365543   77  56  62  58  62  56  53  56  62  58  62  56  77
%e A365543 Row n = 6 counts the following partitions:
%e A365543   (6)       (51)      (42)      (33)      (42)      (51)      (6)
%e A365543   (51)      (411)     (411)     (321)     (411)     (411)     (51)
%e A365543   (42)      (321)     (321)     (3111)    (321)     (321)     (42)
%e A365543   (411)     (3111)    (3111)    (2211)    (3111)    (3111)    (411)
%e A365543   (33)      (2211)    (222)     (21111)   (222)     (2211)    (33)
%e A365543   (321)     (21111)   (2211)    (111111)  (2211)    (21111)   (321)
%e A365543   (3111)    (111111)  (21111)             (21111)   (111111)  (3111)
%e A365543   (222)               (111111)            (111111)            (222)
%e A365543   (2211)                                                      (2211)
%e A365543   (21111)                                                     (21111)
%e A365543   (111111)                                                    (111111)
%t A365543 Table[Length[Select[IntegerPartitions[n],MemberQ[Total/@Subsets[#],k]&]],{n,0,15},{k,0,n}]
%Y A365543 Columns k = 0 and k = n are A000041.
%Y A365543 Central column n = 2k is A002219.
%Y A365543 The complement is counted by A046663, strict A365663.
%Y A365543 Row sums are A304792.
%Y A365543 For subsets instead of partitions we have A365381.
%Y A365543 The strict case is A365661.
%Y A365543 A000009 counts subsets summing to n.
%Y A365543 A000124 counts distinct possible sums of subsets of {1..n}.
%Y A365543 A364272 counts sum-full strict partitions, sum-free A364349.
%Y A365543 Cf. A088809, A093971, A122768, A108917, A299701, A364911, A365541, A365658.
%K A365543 nonn,tabl
%O A365543 0,4
%A A365543 _Gus Wiseman_, Sep 16 2023