A364536 Number of strict integer partitions of n where some part is a difference of two consecutive parts.

0, 0, 0, 1, 0, 0, 2, 1, 2, 2, 5, 4, 6, 6, 9, 11, 16, 17, 23, 25, 30, 38, 48, 55, 65, 78, 92, 106, 127, 146, 176, 205, 230, 277, 315, 366, 421, 483, 552, 640, 727, 829, 950, 1083, 1218, 1408, 1577, 1794, 2017, 2298, 2561, 2919, 3255, 3685, 4116, 4638, 5163

Offset: 0

Author

Gus Wiseman, Jul 31 2023

Keywords

• nonn

Comments

In other words, strict partitions with parts not disjoint from first differences.

Examples

			The a(3) = 1 through a(15) = 11 partitions (A = 10, B = 11, C = 12):
  21  .  .  42   421  431  63   532   542   84    742   743   A5
            321       521  621  541   632   642   841   752   843
                                631   821   651   A21   761   942
                                721   5321  921   5431  842   C21
                                4321        5421  6421  B21   6432
                                            6321  7321  6431  6531
                                                        6521  7431
                                                        7421  7521
                                                        8321  8421
                                                              9321
                                                              54321
		

Crossrefs

For all differences of pairs we have A363226, non-strict A363225.

For all non-differences of pairs we have A364346, strict A364345.

The strict complement is counted by A364464, non-strict A363260.

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

The non-strict case is A364467, ranks A364537.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A323092 counts double-free partitions, strict A120641.

A325325 counts partitions with distinct first-differences, strict A320347.

Cf. A002865, A025065, A093971, A108917, A229816, A236912, A237113, A237667, A326083, A364347.

Programs

Mathematica

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

Python

from collections import Counter
from sympy.utilities.iterables import partitions
def A364536(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 not set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023