A363260 - OEIS

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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

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..50.

EXAMPLE

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)

MATHEMATICA

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

PROG

(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

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.

Sequence in context: A364345 A239455 A362610 * A195012 A333192 A323092

Adjacent sequences: A363257 A363258 A363259 * A363261 A363262 A363263

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 19 2023

STATUS

approved