A364467 Number of integer partitions of n where some part is the difference of two consecutive parts.

0, 0, 0, 1, 1, 2, 4, 5, 9, 13, 21, 28, 42, 55, 78, 106, 144, 187, 255, 325, 429, 554, 717, 906, 1165, 1460, 1853, 2308, 2899, 3582, 4468, 5489, 6779, 8291, 10173, 12363, 15079, 18247, 22124, 26645, 32147, 38555, 46285, 55310, 66093, 78684, 93674, 111104

Offset: 0

Author

Gus Wiseman, Jul 31 2023

Keywords

• nonn

Comments

In other words, the parts are not disjoint from their own first differences.

Examples

			The a(3) = 1 through a(9) = 13 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (431)      (621)
                       (2211)   (3211)    (521)      (3321)
                       (21111)  (22111)   (3221)     (4221)
                                (211111)  (4211)     (4311)
                                          (22211)    (5211)
                                          (32111)    (22221)
                                          (221111)   (32211)
                                          (2111111)  (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)
		

Crossrefs

For all differences of pairs parts we have A363225, complement A364345.

The complement is counted by A363260.

For subsets of {1..n} we have A364466, complement A364463.

The strict case is A364536, complement A364464.

These partitions have ranks A364537.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A050291 counts double-free subsets, complement A088808.

A323092 counts double-free partitions, ranks A320340.

A325325 counts partitions with distinct first differences.

Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.

Programs

Mathematica

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

Python

from collections import Counter
from sympy.utilities.iterables import partitions
def A364467(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), partitions(n,size=True)) if not set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023