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A363994
a(n) is the number of partitions of n whose difference multiset has no duplicates; see Comments.
3
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}.
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Sep 08 2023
EXTENSIONS
More terms from Alois P. Heinz, Sep 12 2023
STATUS
approved