A363994 -id:A363994 - cp's OEIS frontend

A364612 a(n) = number of partitions of n whose difference multiset has at least one duplicate; see Comments.

0, 0, 0, 1, 2, 4, 7, 10, 15, 24, 32, 45, 66, 86, 117, 158, 206, 268, 357, 452, 583, 745, 948, 1188, 1507, 1874, 2348, 2908, 3604, 4428, 5472, 6675, 8169, 9939, 12096, 14622, 17713, 21322, 25687, 30808, 36924, 44107, 52701, 62697, 74572, 88457, 104850, 123934

Offset: 0

Clark Kimberling, Sep 08 2023

nonn

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

			The partitions of 8 are [8], [7,1], [6,2], [6,1,1], [5,3], [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 15 partitions whose difference multiset includes at least one duplicate are all the 22 partitions of 8 except these 7: [8], [7,1], [6,2], [5,3], [6,2], [5,3], [5,2,1], [4,3,1].

Cf. A000041, A363994.

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], False];
Table[a[n], {n, 1, 30}]

from collections import Counter
from itertools import combinations
from sympy.utilities.iterables import partitions
def A364612(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

a(n) = A000041(n)-A364994(n).

More terms from Alois P. Heinz, Sep 12 2023

A364613 a(n) = number of partitions of n whose sum multiset is free of duplicates; see Comments.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 18, 20, 26, 29, 36, 38, 50, 53, 67, 69, 89, 95, 115, 122, 151, 161, 195, 201, 247, 266, 312, 330, 386, 419, 487, 520, 600, 641, 742, 793, 901, 979, 1088, 1186, 1331, 1454, 1605, 1730, 1925, 2102, 2311, 2525, 2741, 3001

Offset: 0

Clark Kimberling, Sep 17 2023

nonn

If M is a multiset of real numbers, then the sum multiset of M consists of the sums of pairs of distinct numbers in M. For example, the sum multiset of (1,2,4,5) is {3,5,6,6,7,9}.

			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 sum multiset is duplicate-free are [8], [7,1], [6,2], [5,3], [5,2,1], [4,4], [4,3,1].

Cf. A000041, A236912, A325877, A363994.

s[n_, k_] := s[n, k] = Subsets[IntegerPartitions[n][[k]], {2}];
g[n_, k_] := g[n, k] = DuplicateFreeQ[Map[Total, s[n, k]]];
t[n_] := Table[g[n, k], {k, 1, PartitionsP[n]}];
a[n_] := Count[t[n], True]
Table[a[n], {n, 1, 40}]

a(n) = A325877(n) - (1 - n mod 2) for n > 0. - Andrew Howroyd, Sep 17 2023

More terms from Alois P. Heinz, Sep 17 2023

cp's OEIS Frontend

A364612 a(n) = number of partitions of n whose difference multiset has at least one duplicate; see Comments.

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