A363994 - OEIS

%I #19 Sep 18 2023 02:10:41

%S 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,

%T 88,102,114,137,132,167,180,204,214,261,264,315,328,377,414,476,473,

%U 564,603,677,708,820,846,969,1028,1131,1214,1364,1414,1596,1701,1858

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

%C 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}.

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

%F a(n) = A325876(n) - (1 - n mod 2) for n > 0. - _Andrew Howroyd_, Sep 17 2023

%e 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].

%e 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].

%t s[n_, k_] := s[n, k] = Subsets[IntegerPartitions[n][[k]], {2}]

%t g[n_, k_] := g[n, k] = DuplicateFreeQ[Map[Differences, s[n, k]]]

%t t[n_] := t[n] = Table[g[n, k], {k, 1, PartitionsP[n]}];

%t a[n_] := Count[t[n], True];

%t Table[a[n], {n, 1, 20}]

%o (Python)

%o from collections import Counter

%o from itertools import combinations

%o from sympy.utilities.iterables import partitions

%o 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

%Y Cf. A000041, A325858, A325876, A364612.

%K nonn

%O 0,3

%A _Clark Kimberling_, Sep 08 2023

%E More terms from _Alois P. Heinz_, Sep 12 2023