A363994 a(n) is the number of partitions of n whose difference multiset has no duplicates; see Comments.

1, 1, 2, 2, 3, 3, 4, 5, 7, 6, 10, 11, 11, 15, 18, 18, 25, 29, 28, 38, 44, 47, 54, 67, 68, 84, 88, 102, 114, 137, 132, 167, 180, 204, 214, 261, 264, 315, 328, 377, 414, 476, 473, 564, 603, 677, 708, 820, 846, 969, 1028, 1131, 1214, 1364, 1414, 1596, 1701, 1858

Offset: 0

Author

Clark Kimberling, Sep 08 2023

Keywords

• nonn

Comments

If M is a multiset of real numbers, then the difference multiset of M consists of the differences of pairs of numbers in M. For example, the difference multiset of {1,2,2,5} is {0,1,1,3,3,4}.

Examples

			The partitions of 8 are [8], [7,1], [6,2], [6,1,1], [5,3], [5,2,1], [5,1,1,1], [4,4], [4,3,1], [4,2,2], [4,2,1,1], [4,1,1,1,1], [3,3,2], [3,3,1,1], [3,2,2,1], [3,2,1,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1].
The 7 partitions whose difference multiset is duplicate-free are [8], [7,1], [6,2], [5,3], [5,2,1], [4,4], [4,3,1].
		

Crossrefs

Cf. A000041, A325858, A325876, A364612.

Programs

Mathematica

s[n_, k_] := s[n, k] = Subsets[IntegerPartitions[n][[k]], {2}]
g[n_, k_] := g[n, k] = DuplicateFreeQ[Map[Differences, s[n, k]]]
t[n_] := t[n] = Table[g[n, k], {k, 1, PartitionsP[n]}];
a[n_] := Count[t[n], True];
Table[a[n], {n, 1, 20}]

Python

from collections import Counter
from itertools import combinations
from sympy.utilities.iterables import partitions
def A363994(n): return sum(1 for p in partitions(n) if max(list(Counter(abs(d[0]-d[1]) for d in combinations(list(Counter(p).elements()),2)).values()),default=1)==1) # Chai Wah Wu, Sep 17 2023

Formula

a(n) = A000041(n) - A364612(n).

a(n) = A325876(n) - (1 - n mod 2) for n > 0. - Andrew Howroyd, Sep 17 2023

Extensions

More terms from Alois P. Heinz, Sep 12 2023