A365659 Number of strict integer partitions of n that either have (1) length 2, or (2) greatest part n/2.

0, 0, 0, 1, 1, 2, 3, 3, 4, 4, 6, 5, 8, 6, 10, 7, 12, 8, 15, 9, 18, 10, 21, 11, 25, 12, 29, 13, 34, 14, 40, 15, 46, 16, 53, 17, 62, 18, 71, 19, 82, 20, 95, 21, 109, 22, 125, 23, 144, 24, 165, 25, 189, 26, 217, 27, 248, 28, 283, 29, 324

Offset: 0

Author

Gus Wiseman, Sep 16 2023

Keywords

• nonn

Comments

Also the number of strict integer partitions of n containing two possibly equal elements summing to n.

Examples

			The a(3) = 1 through a(11) = 5 partitions:
  (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)    (5,4)  (6,4)    (6,5)
                (4,1)  (5,1)    (5,2)  (6,2)    (6,3)  (7,3)    (7,4)
                       (3,2,1)  (6,1)  (7,1)    (7,2)  (8,2)    (8,3)
                                       (4,3,1)  (8,1)  (9,1)    (9,2)
                                                       (5,3,2)  (10,1)
                                                       (5,4,1)
		

Crossrefs

Without repeated parts we have A140106.

The non-strict version is A238628.

For subsets instead of strict partitions we have A365544.

A000009 counts subsets summing to n.

A365046 counts combination-full subsets, differences of A364914.

A365543 counts partitions of n with a submultiset summing to k.

Cf. A008967, A046663, A068911, A095944, A364272, A365376, A365377.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(Length[#]==2||Max@@#==n/2)&]], {n,0,30}]

Python

from sympy.utilities.iterables import partitions
def A365659(n): return n>>1 if n&1 or n==0 else (m:=n>>1)+sum(1 for p in partitions(m) if max(p.values(),default=1)==1)-2 # Chai Wah Wu, Sep 18 2023

Formula

a(n) = (n-1)/2 if n is odd. a(n) = n/2 + A000009(n/2) - 2 if n is even and n > 0. - Chai Wah Wu, Sep 18 2023