A364464 Number of strict integer partitions of n where no part is the difference of two consecutive parts.

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

Offset: 0

Author

Gus Wiseman, Jul 30 2023

Keywords

• nonn

Comments

In other words, the parts are disjoint from the first differences.

Examples

			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)
		

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.

The non-strict version is A363260.

The complement is counted by A364536, non-strict A364467.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A120641 counts strict double-free partitions, non-strict A323092.

A320347 counts strict partitions w/ distinct differences, non-strict A325325.

Cf. A002865, A025065, A236912, A237667, A275972, A363226, A364345, A364465.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]=={}&]],{n,0,15}]

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