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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.

0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 25, 38, 46, 64, 76, 106, 124, 167, 199, 261, 309, 402, 471, 604, 714, 898, 1053, 1323, 1542, 1911, 2237, 2745, 3201, 3913, 4536, 5506, 6402, 7706, 8918, 10719, 12364, 14760, 17045, 20234, 23296, 27600, 31678, 37365, 42910, 50371, 57695, 67628, 77300, 90242, 103131, 119997

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OFFSET

0,5

COMMENTS

The complement (complete partitions) isA126796.

LINKS

Joerg Arndt,Table of n, a(n) for n = 0..10000

FORMULA

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

EXAMPLE

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

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

(2,2) (3,2) (3,3) (4,3) (4,4)

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

(5,1) (6,1) (6,2)

(2,2,2) (3,2,2) (7,1)

(4,1,1) (3,3,1) (3,3,2)

(5,1,1) (4,2,2)

(4,3,1)

(5,2,1)

(6,1,1)

(2,2,2,2)

(5,1,1,1)

MATHEMATICA

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

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

CROSSREFS

For parts instead of sums we haveA047967/A365919, ranksA080259/A055932.

The complement isA126796, ranksA325781, strictA188431.

These partitions have ranksA365830.

The strict case isA365831.

Row sums ofA365923 without the first column, strictA365545.

A000041 counts integer partitions, strictA000009.

A046663 counts partitions w/o a submultiset summing to k, strictA365663.

A276024 counts positive subset-sums of partitions, strictA284640.

A325799 counts non-subset-sums of prime indices.

A364350 counts combination-free strict partitions.

A365543 counts partitions with a submultiset summing to k, strictA365661.

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

Sequence in context:A240449 A088571 A325834 *A241832 A027187 A056508

Adjacent sequences:A365921 A365922 A365923 *A365925 A365926 A365927

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 26 2023

STATUS

approved