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A364467
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Number of integer partitions of n where some part is the difference of two consecutive parts.
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12
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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
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OFFSET
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0,6
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COMMENTS
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In other words, the parts are not disjoint from their own first differences.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Intersection[#, -Differences[#]]!={}&]], {n, 0, 30}]
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PROG
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(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
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CROSSREFS
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For all differences of pairs parts we have A363225, complement A364345.
The complement is counted by A363260.
These partitions have ranks A364537.
A325325 counts partitions with distinct first differences.
Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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