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Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.
29

%I #14 Sep 28 2023 04:41:17

%S 0,0,1,1,3,3,6,7,12,14,22,25,38,46,64,76,106,124,167,199,261,309,402,

%T 471,604,714,898,1053,1323,1542,1911,2237,2745,3201,3913,4536,5506,

%U 6402,7706,8918,10719,12364,14760,17045,20234,23296,27600,31678,37365,42910,50371,57695,67628,77300,90242,103131,119997

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

%C The complement (complete partitions) is A126796.

%H Joerg Arndt, <a href="/A365924/b365924.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A000041(n) - A126796(n).

%e The a(0) = 0 through a(8) = 12 partitions:

%e . . (2) (3) (4) (5) (6) (7) (8)

%e (2,2) (3,2) (3,3) (4,3) (4,4)

%e (3,1) (4,1) (4,2) (5,2) (5,3)

%e (5,1) (6,1) (6,2)

%e (2,2,2) (3,2,2) (7,1)

%e (4,1,1) (3,3,1) (3,3,2)

%e (5,1,1) (4,2,2)

%e (4,3,1)

%e (5,2,1)

%e (6,1,1)

%e (2,2,2,2)

%e (5,1,1,1)

%t nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];

%t Table[Length[Select[IntegerPartitions[n],Length[nmz[#]]>0&]],{n,0,15}]

%Y For parts instead of sums we have A047967/A365919, ranks A080259/A055932.

%Y The complement is A126796, ranks A325781, strict A188431.

%Y These partitions have ranks A365830.

%Y The strict case is A365831.

%Y Row sums of A365923 without the first column, strict A365545.

%Y A000041 counts integer partitions, strict A000009.

%Y A046663 counts partitions w/o a submultiset summing to k, strict A365663.

%Y A276024 counts positive subset-sums of partitions, strict A284640.

%Y A325799 counts non-subset-sums of prime indices.

%Y A364350 counts combination-free strict partitions.

%Y A365543 counts partitions with a submultiset summing to k, strict A365661.

%Y Cf. A002865, A006827, A018818, A264401, A299701, A304792, A364272, A365658, A365918, A365921.

%K nonn

%O 0,5

%A _Gus Wiseman_, Sep 26 2023