A365002 Number of ways to write n as a nonnegative linear combination of a strict integer partition.

1, 1, 2, 4, 8, 10, 26, 32, 63, 84, 157, 207, 383, 477, 768, 1108, 1710, 2261, 3536, 4605, 6869, 9339, 13343, 17653, 25785, 33463, 46752, 61549, 85614, 110861, 153719, 197345, 268623, 346845, 463513, 593363, 797082, 1011403, 1335625, 1703143, 2232161, 2820539

Offset: 0

Author

Gus Wiseman, Aug 22 2023

Keywords

• nonn

Comments

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

Examples

			The a(1) = 1 through a(5) = 10 ways:
  1*1  1*2  1*3      1*4      1*5
       2*1  3*1      2*2      5*1
            0*2+3*1  4*1      0*2+5*1
            1*2+1*1  0*2+4*1  0*3+5*1
                     0*3+4*1  0*4+5*1
                     1*2+2*1  1*2+3*1
                     1*3+1*1  1*3+1*2
                     2*2+0*1  1*3+2*1
                              1*4+1*1
                              2*2+1*1
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Row sums of lower-left half of A364916 as an array.

Column sums of right half of A364916 as a triangle.

For all positive coefficients we have A000041, non-strict A006951.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A364350 counts combination-free strict partitions, complement A364839.

A364913 counts combination-full partitions.

Cf. A066328, A116861, A364272, A364907, A364910, A364911, A364912, A365004.

Programs

Mathematica

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Sum[Length[combs[n,y]], {y,Select[Join@@IntegerPartitions/@Range[n], UnsameQ@@#&]}],{n,0,15}]

Python

from itertools import combinations
from collections import Counter
from sympy.utilities.iterables import partitions
def A365002(n):
    aset = Counter(tuple(sorted(set(p))) for p in partitions(n))
    return sum(sum(aset[t] for t in aset if set(t).issubset(set(q))) for l in range(1,n+1) for q in combinations(range(1,n+1),l) if sum(q)<=n) # Chai Wah Wu, Sep 20 2023

Extensions

a(16)-a(34) from Chai Wah Wu, Sep 20 2023

a(35)-a(38) from Chai Wah Wu, Sep 21 2023

a(0)=1 and a(39)-a(41) from Alois P. Heinz, Jan 11 2024