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

0,3

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}.

Links

Table of n, a(n) for n=0..57.

Formula

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

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

Example

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].

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}]

Prog

(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

Crossrefs

Cf. A000041, A325858, A325876, A364612.

Sequence in context: A174619 A130083 A369787 * A286219 A117357 A029020

Adjacent sequences: A363991 A363992 A363993 * A363995 A363996 A363997

Keyword

nonn

Author

Clark Kimberling, Sep 08 2023

Extensions

More terms from Alois P. Heinz, Sep 12 2023

Status

approved