A276107 Number of partitions of n which can themselves be subdivided into two partitions whose sums differ by 1 at most.

1, 1, 1, 2, 3, 5, 6, 11, 14, 22, 25, 43, 53, 79, 89, 140, 167, 243, 278, 409, 480, 666, 760, 1082, 1273, 1708, 1948, 2649, 3089, 4073, 4682, 6180, 7177, 9213, 10565, 13660, 15869, 19987, 22911, 29012, 33601, 41762, 47942, 59571, 68756, 84240, 96570, 118641

Offset

0,4

Comments

Number of partitions whose summands can be divided into two sets whose sums are equal, or whose sums differ by one.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..115 (first 77 terms from Chai Wah Wu)

Formula

a(n) = A000041(n) - A046663(n,floor(n/2)). - Max Alekseyev, Oct 17 2022

Example

For example: the partition of 6 into 2+2+2 is not counted because no subset of {2,2,2} has the sum 3. The partition of 6 into 3+2+1 is counted because {3,2,1} can be divided into two subsets {3} and {2,1} each of which has sum 3. The partition of 7 into 4+2+1 is counted because {4,2,1} can be broken up into sets {4} and {2,1} the sum of whose elements differ by one.

Maple

N:= 50: # to get a(0) .. a(N)
Pm1:= combinat:-partition(0);
for m from 0 to N/2 do
  Pm:= Pm1;
  A[2*m]:= nops({seq(seq(sort([op(Pm[i]), op(Pm[j])]), j=1..i), i=1..nops(Pm))});
  if 2*m+1 > N then break fi;
  Pm1:= combinat:-partition(m+1);
  A[2*m+1]:= nops({seq(seq(sort([op(Pm[i]), op(Pm1[j])]), j=1..nops(Pm1)), i=1..nops(Pm))});
od:
seq(A[i], i=  0..N); # Robert Israel, Aug 18 2016

Prog

(Python)
from collections import Counter
from sympy.utilities.iterables import partitions
def A276107(n): return len({tuple(sorted((p+q).items())) for p in (Counter(p) for p in partitions(n>>1)) for q in (Counter(q) for q in partitions(n+1>>1))}) # Chai Wah Wu, Sep 20 2023

Crossrefs

Cf. A000041, A002219, A046663.

Sequence in context: A240490 A039839 A039844 * A187068 A187070 A038196

Adjacent sequences: A276104 A276105 A276106 * A276108 A276109 A276110

Keyword

nonn

Author

David S. Newman, Aug 18 2016

Extensions

a(11)-a(47) from Alois P. Heinz, Aug 18 2016

Status

approved