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A364673
Number of (necessarily strict) integer partitions of n containing all of their own first differences.
9
1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 2, 5, 2, 2, 4, 2, 3, 6, 4, 4, 8, 4, 4, 10, 8, 7, 8, 13, 9, 15, 12, 13, 17, 20, 15, 31, 24, 27, 32, 33, 32, 50, 42, 45, 53, 61, 61, 85, 76, 86, 101, 108, 118, 137, 141, 147, 179, 184, 196, 222, 244, 257, 295, 324, 348, 380, 433
OFFSET
0,4
LINKS
EXAMPLE
The partition y = (12,6,3,2,1) has differences (6,3,1,1), and {1,3,6} is a subset of {1,2,3,6,12}, so y is counted under a(24).
The a(n) partitions for n = 1, 3, 6, 12, 15, 18, 21:
(1) (3) (6) (12) (15) (18) (21)
(2,1) (4,2) (8,4) (10,5) (12,6) (14,7)
(3,2,1) (6,4,2) (8,4,2,1) (9,6,3) (12,6,3)
(5,4,2,1) (5,4,3,2,1) (6,5,4,2,1) (8,6,4,2,1)
(6,3,2,1) (7,5,3,2,1) (9,5,4,2,1)
(8,4,3,2,1) (9,6,3,2,1)
(10,5,3,2,1)
(6,5,4,3,2,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&SubsetQ[#, -Differences[#]]&]], {n, 0, 30}]
PROG
(Python)
from collections import Counter
def A364673_list(maxn):
count = Counter()
for i in range(maxn//3):
A, f, i = [[(i+1, )]], 0, 0
while f == 0:
A.append([])
for j in A[i]:
for k in j:
x = j + (j[-1] + k, )
y = sum(x)
if y <= maxn:
A[i+1].append(x)
count.update({y})
if len(A[i+1]) < 1: f += 1
i += 1
return [count[z]+1 for z in range(maxn+1)] # John Tyler Rascoe, Mar 09 2024
CROSSREFS
Containing all differences: A007862.
Containing no differences: A364464, strict complement A364536.
Containing at least one difference: A364467, complement A363260.
For subsets of {1..n} we have A364671, complement A364672.
A non-strict version is A364674.
For submultisets instead of subsets we have A364675.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions w/o re-used parts, complement A237113.
A325325 counts partitions with distinct first differences.
Sequence in context: A165162 A125106 A229874 * A330439 A243611 A273102
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2023
STATUS
approved