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Number of integer partitions of n where some part is the difference of two consecutive parts.
12

%I #10 Sep 26 2023 13:39:20

%S 0,0,0,1,1,2,4,5,9,13,21,28,42,55,78,106,144,187,255,325,429,554,717,

%T 906,1165,1460,1853,2308,2899,3582,4468,5489,6779,8291,10173,12363,

%U 15079,18247,22124,26645,32147,38555,46285,55310,66093,78684,93674,111104

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

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

%e The a(3) = 1 through a(9) = 13 partitions:

%e (21) (211) (221) (42) (421) (422) (63)

%e (2111) (321) (2221) (431) (621)

%e (2211) (3211) (521) (3321)

%e (21111) (22111) (3221) (4221)

%e (211111) (4211) (4311)

%e (22211) (5211)

%e (32111) (22221)

%e (221111) (32211)

%e (2111111) (42111)

%e (222111)

%e (321111)

%e (2211111)

%e (21111111)

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

%o (Python)

%o from collections import Counter

%o from sympy.utilities.iterables import partitions

%o 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

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

%Y The complement is counted by A363260.

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

%Y The strict case is A364536, complement A364464.

%Y These partitions have ranks A364537.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A050291 counts double-free subsets, complement A088808.

%Y A323092 counts double-free partitions, ranks A320340.

%Y A325325 counts partitions with distinct first differences.

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

%K nonn

%O 0,6

%A _Gus Wiseman_, Jul 31 2023