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A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

%I A365924 #14 Sep 28 2023 04:41:17
%S A365924 0,0,1,1,3,3,6,7,12,14,22,25,38,46,64,76,106,124,167,199,261,309,402,
%T A365924 471,604,714,898,1053,1323,1542,1911,2237,2745,3201,3913,4536,5506,
%U A365924 6402,7706,8918,10719,12364,14760,17045,20234,23296,27600,31678,37365,42910,50371,57695,67628,77300,90242,103131,119997
%N A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.
%C A365924 The complement (complete partitions) is A126796.
%H A365924 Joerg Arndt, <a href="/A365924/b365924.txt">Table of n, a(n) for n = 0..10000</a>
%F A365924 a(n) = A000041(n) - A126796(n).
%e A365924 The a(0) = 0 through a(8) = 12 partitions:
%e A365924   .  .  (2)  (3)  (4)    (5)    (6)      (7)      (8)
%e A365924                   (2,2)  (3,2)  (3,3)    (4,3)    (4,4)
%e A365924                   (3,1)  (4,1)  (4,2)    (5,2)    (5,3)
%e A365924                                 (5,1)    (6,1)    (6,2)
%e A365924                                 (2,2,2)  (3,2,2)  (7,1)
%e A365924                                 (4,1,1)  (3,3,1)  (3,3,2)
%e A365924                                          (5,1,1)  (4,2,2)
%e A365924                                                   (4,3,1)
%e A365924                                                   (5,2,1)
%e A365924                                                   (6,1,1)
%e A365924                                                   (2,2,2,2)
%e A365924                                                   (5,1,1,1)
%t A365924 nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
%t A365924 Table[Length[Select[IntegerPartitions[n],Length[nmz[#]]>0&]],{n,0,15}]
%Y A365924 For parts instead of sums we have A047967/A365919, ranks A080259/A055932.
%Y A365924 The complement is A126796, ranks A325781, strict A188431.
%Y A365924 These partitions have ranks A365830.
%Y A365924 The strict case is A365831.
%Y A365924 Row sums of A365923 without the first column, strict A365545.
%Y A365924 A000041 counts integer partitions, strict A000009.
%Y A365924 A046663 counts partitions w/o a submultiset summing to k, strict A365663.
%Y A365924 A276024 counts positive subset-sums of partitions, strict A284640.
%Y A365924 A325799 counts non-subset-sums of prime indices.
%Y A365924 A364350 counts combination-free strict partitions.
%Y A365924 A365543 counts partitions with a submultiset summing to k, strict A365661.
%Y A365924 Cf. A002865, A006827, A018818, A264401, A299701, A304792, A364272, A365658, A365918, A365921.
%K A365924 nonn
%O A365924 0,5
%A A365924 _Gus Wiseman_, Sep 26 2023