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A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

%I A365921 #11 Dec 13 2024 09:41:56
%S A365921 1,1,0,1,1,0,2,0,1,0,2,0,1,2,0,4,0,0,1,2,0,5,0,0,1,1,4,0,8,0,0,0,1,2,
%T A365921 4,0,10,0,0,0,2,1,2,7,0,16,0,0,0,0,2,1,3,8,0,20,0,0,0,0,2,2,2,4,12,0,
%U A365921 31,0,0,0,0,0,2,2,2,5,14,0
%N A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.
%H A365921 Steven R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009.
%e A365921 The partition (6,2,1,1) has subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 so is counted under T(10,5).
%e A365921 Triangle begins:
%e A365921    1
%e A365921    1  0
%e A365921    1  1  0
%e A365921    2  0  1  0
%e A365921    2  0  1  2  0
%e A365921    4  0  0  1  2  0
%e A365921    5  0  0  1  1  4  0
%e A365921    8  0  0  0  1  2  4  0
%e A365921   10  0  0  0  2  1  2  7  0
%e A365921   16  0  0  0  0  2  1  3  8  0
%e A365921   20  0  0  0  0  2  2  2  4 12  0
%e A365921   31  0  0  0  0  0  2  2  2  5 14  0
%e A365921   39  0  0  0  0  0  4  2  2  3  6 21  0
%e A365921   55  0  0  0  0  0  0  4  2  4  3  9 24  0
%e A365921   71  0  0  0  0  0  0  5  4  2  4  5 10 34  0
%e A365921 Row n = 8 counts the following partitions:
%e A365921   (4211)      .  .  .  (521)   (611)  (71)   (8)     .
%e A365921   (41111)              (5111)         (431)  (62)
%e A365921   (3311)                                     (53)
%e A365921   (3221)                                     (44)
%e A365921   (32111)                                    (422)
%e A365921   (311111)                                   (332)
%e A365921   (22211)                                    (2222)
%e A365921   (221111)
%e A365921   (2111111)
%e A365921   (11111111)
%t A365921 nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
%t A365921 Table[Length[Select[IntegerPartitions[n],Max@@Prepend[nmz[#],0]==k&]],{n,0,10},{k,0,n}]
%Y A365921 Row sums are A000041.
%Y A365921 Diagonal k = n-1 is A002865.
%Y A365921 Column k = 1 is A126796 (complete partitions), ranks A325781.
%Y A365921 Central diagonal n = 2k is A126796 also.
%Y A365921 For parts instead of sums we have A339737, rank stat A339662, min A257993.
%Y A365921 This is the triangle for the rank statistic A365920.
%Y A365921 Latter row sums are A365924 (incomplete partitions), ranks A365830.
%Y A365921 Column sums are A366127.
%Y A365921 A055932 lists numbers whose prime indices cover an initial interval.
%Y A365921 A056239 adds up prime indices, row sums of A112798.
%Y A365921 A073491 lists numbers with gap-free prime indices.
%Y A365921 A238709/A238710 count partitions by least/greatest difference.
%Y A365921 A342050/A342051 have prime indices with odd/even least gap.
%Y A365921 A366128 gives the least non-subset-sum of prime indices.
%Y A365921 Cf. A001522, A079068, A098743, A264401, A286469 or A286470, A339886.
%K A365921 nonn,tabl
%O A365921 0,7
%A A365921 _Gus Wiseman_, Sep 30 2023