A325876 -id:A325876 - cp's OEIS frontend

A325875 Number of compositions of n whose differences of all degrees > 1 are nonzero.

1, 1, 2, 3, 7, 13, 20, 38, 69, 129, 222, 407, 726, 1313, 2318, 4146, 7432, 13296, 23759, 42458, 75714

Offset: 0

Gus Wiseman, Jun 02 2019

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.
A composition of n is a finite sequence of positive integers with sum n.
The case for all degrees including 1 is A325851.

			The a(1) = 1 through a(6) = 20 compositions:
  (1)  (2)   (3)   (4)    (5)     (6)
       (11)  (12)  (13)   (14)    (15)
             (21)  (22)   (23)    (24)
                   (31)   (32)    (33)
                   (112)  (41)    (42)
                   (121)  (113)   (51)
                   (211)  (122)   (114)
                          (131)   (132)
                          (212)   (141)
                          (221)   (213)
                          (311)   (231)
                          (1121)  (312)
                          (1211)  (411)
                                  (1122)
                                  (1131)
                                  (1212)
                                  (1311)
                                  (2121)
                                  (2211)
                                  (11211)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325874, A325876.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,2,Length[#]}],0]&]],{n,0,10}]

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

Original entry on oeis.org

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

Clark Kimberling, Sep 08 2023

nonn

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}.

			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].

Cf. A000041, A325858, A325876, A364612.

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}]

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

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

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

More terms from Alois P. Heinz, Sep 12 2023

cp's OEIS Frontend

A325875 Number of compositions of n whose differences of all degrees > 1 are nonzero.

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