A363260 Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.

1, 1, 2, 2, 4, 5, 7, 10, 13, 17, 21, 28, 35, 46, 57, 70, 87, 110, 130, 165, 198, 238, 285, 349, 410, 498, 583, 702, 819, 983, 1136, 1353, 1570, 1852, 2137, 2520, 2898, 3390, 3891, 4540, 5191, 6028, 6889, 7951, 9082, 10450, 11884, 13650, 15508, 17728, 20113

Offset: 0

Author

Gus Wiseman, Jul 19 2023

Keywords

• nonn

Examples

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (611)
                                               (4111)     (2222)
                                               (31111)    (3311)
                                               (1111111)  (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)
		

Crossrefs

For length instead of differences we have A229816, strict A240861.

For all differences of pairs parts we have A364345.

For subsets of {1..n} instead of partitions we have A364463.

The strict case is A364464.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A323092 counts double-free partitions, ranks A320340.

A325325 counts partitions with distinct first-differences.

Cf. A002865, A025065, A108917, A236912, A237113, A237667, A320347, A326083, A363225, A364347.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]=={}&]],{n,0,30}]

Python

from collections import Counter
from sympy.utilities.iterables import partitions
def A363260(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), 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