A365827 Number of strict integer partitions of n whose length is not 2.

1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 25, 30, 38, 45, 55, 66, 79, 93, 111, 130, 153, 179, 209, 242, 282, 325, 375, 432, 496, 568, 651, 742, 846, 963, 1094, 1240, 1406, 1589, 1795, 2026, 2282, 2567, 2887, 3240, 3634, 4072, 4557, 5094, 5692, 6351

Offset

0,7

Comments

Also the number of strict integer partitions of n with no pair of distinct parts summing to n.

Links

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

Formula

a(n) = A000009(n) - A004526(n-1) for n > 0.

Example

The a(5) = 1 through a(13) = 12 strict partitions (A..D = 10..13):
  (5)  (6)    (7)    (8)    (9)    (A)     (B)     (C)     (D)
       (321)  (421)  (431)  (432)  (532)   (542)   (543)   (643)
                     (521)  (531)  (541)   (632)   (642)   (652)
                            (621)  (631)   (641)   (651)   (742)
                                   (721)   (731)   (732)   (751)
                                   (4321)  (821)   (741)   (832)
                                           (5321)  (831)   (841)
                                                   (921)   (931)
                                                   (5421)  (A21)
                                                   (6321)  (5431)
                                                           (6421)
                                                           (7321)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[#]!=2&]], {n, 0, 30}]

Crossrefs

The complement is counted by A140106 shifted left.

Heinz numbers are A005117 \ A006881 = A005117 /\ A100959.

The non-strict version is A058984, complement A004526.

The case not containing n/2 is A365826, non-strict A365825.

A000041 counts integer partitions, strict A000009.

A046663 counts partitions with no submultiset summing to k, strict A365663.

A182616 counts partitions of 2n that do not contain n, strict A365828.

Cf. A008967, A035363, A078408, A365659.

Sequence in context: A102464 A082538 A035939 * A116665 A122135 A339572

Adjacent sequences: A365824 A365825 A365826 * A365828 A365829 A365830

Keyword

nonn

Author

Gus Wiseman, Sep 20 2023

Status

approved