|
|
A364464
|
|
Number of strict integer partitions of n where no part is the difference of two consecutive parts.
|
|
14
|
|
|
1, 1, 1, 1, 2, 3, 2, 4, 4, 6, 5, 8, 9, 12, 13, 16, 16, 21, 23, 29, 34, 38, 41, 49, 57, 64, 73, 86, 95, 110, 120, 135, 160, 171, 197, 219, 247, 277, 312, 342, 386, 431, 476, 527, 598, 640, 727, 796, 893, 966, 1097, 1178, 1327, 1435, 1602, 1740, 1945, 2084, 2337
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
In other words, the parts are disjoint from the first differences.
|
|
LINKS
|
|
|
EXAMPLE
|
The strict partition y = (9,5,3,1) has differences (4,2,2), and these are disjoint from the parts, so y is counted under a(18).
The a(1) = 1 through a(9) = 6 strict partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(3,1) (3,2) (5,1) (4,3) (5,3) (5,4)
(4,1) (5,2) (6,2) (7,2)
(6,1) (7,1) (8,1)
(4,3,2)
(5,3,1)
|
|
MATHEMATICA
|
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, -Differences[#]]=={}&]], {n, 0, 15}]
|
|
PROG
|
(Python)
from collections import Counter
from sympy.utilities.iterables import partitions
def A364464(n): return sum(1 for s, p in map(lambda x: (x[0], tuple(sorted(Counter(x[1]).elements()))), filter(lambda p:max(p[1].values(), default=1)==1, partitions(n, size=True))) if set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023
|
|
CROSSREFS
|
For length instead of differences we have A240861, non-strict A229816.
For all differences of pairs of elements we have A364346, for subsets A007865.
For subsets instead of strict partitions we have A364463, complement A364466.
A320347 counts strict partitions w/ distinct differences, non-strict A325325.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|