Movatterモバイル変換

A365831

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

0, 0, 1, 1, 2, 3, 3, 4, 6, 8, 9, 11, 13, 16, 21, 25, 31, 36, 43, 50, 59, 69, 82, 96, 113, 131, 155, 179, 208, 239, 276, 315, 362, 414, 472, 539, 614, 698, 795, 902, 1023, 1158, 1311, 1479, 1672, 1881, 2118, 2377, 2671, 2991, 3354, 3748, 4194, 4679, 5223, 5815

(list;graph;refs;listen;history;text;internal format)

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..55.

EXAMPLE

The strict partition (14,5,4,2,1) has no subset summing to 13 so is counted under a(26).

The a(2) = 1 through a(10) = 9 strict partitions:

(2) (3) (4) (5) (6) (7) (8) (9) (10)

(3,1) (3,2) (4,2) (4,3) (5,3) (5,4) (6,4)

(4,1) (5,1) (5,2) (6,2) (6,3) (7,3)

(6,1) (7,1) (7,2) (8,2)

(4,3,1) (8,1) (9,1)

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

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

(6,2,1) (6,3,1)

(7,2,1)

MATHEMATICA

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

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

CROSSREFS

For parts instead of sums we have ranksA080259,A055932.

The strict complement isA188431, non-strictA126796 (ranksA325781).

Row sums ofA365545 without the first column, non-strictA365923.

The non-strict version isA365924, ranksA365830.

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.

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

Cf.A006827,A047967,A299701,A304792,A364350,A365658,A365918,A365921.

Sequence in context:A241447 A081230 A341144 *A036021 A036025 A036030

Adjacent sequences:A365828 A365829 A365830 *A365832 A365833 A365834

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 28 2023

STATUS

approved