The On-Line Encyclopedia of Integer Sequences (OEIS)

A364536 revision #7

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

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OFFSET

0,7

COMMENTS

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

LINKS

Table of n, a(n) for n=0..56.

EXAMPLE

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

MATHEMATICA

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

PROG

(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

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.

Sequence in context: A325110 A357888 A145862 * A058701 A004559 A214968

Adjacent sequences: A364533 A364534 A364535 * A364537 A364538 A364539

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 31 2023

STATUS

proposed