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Number of strict integer partitions of n where no part is the difference of two consecutive parts.
14

%I #12 Sep 26 2023 13:38:59

%S 1,1,1,1,2,3,2,4,4,6,5,8,9,12,13,16,16,21,23,29,34,38,41,49,57,64,73,

%T 86,95,110,120,135,160,171,197,219,247,277,312,342,386,431,476,527,

%U 598,640,727,796,893,966,1097,1178,1327,1435,1602,1740,1945,2084,2337

%N Number of strict integer partitions of n where no part is the difference of two consecutive parts.

%C In other words, the parts are disjoint from the first differences.

%e The strict partition y = (9,5,3,1) has differences (4,2,2), and these are disjoint from the parts, so y is counted under a(18).

%e The a(1) = 1 through a(9) = 6 strict partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8) (9)

%e (3,1) (3,2) (5,1) (4,3) (5,3) (5,4)

%e (4,1) (5,2) (6,2) (7,2)

%e (6,1) (7,1) (8,1)

%e (4,3,2)

%e (5,3,1)

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]=={}&]],{n,0,15}]

%o (Python)

%o from collections import Counter

%o from sympy.utilities.iterables import partitions

%o def A364464(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 set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # _Chai Wah Wu_, Sep 26 2023

%Y For length instead of differences we have A240861, non-strict A229816.

%Y For all differences of pairs of elements we have A364346, for subsets A007865.

%Y For subsets instead of strict partitions we have A364463, complement A364466.

%Y The non-strict version is A363260.

%Y The complement is counted by A364536, non-strict A364467.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A120641 counts strict double-free partitions, non-strict A323092.

%Y A320347 counts strict partitions w/ distinct differences, non-strict A325325.

%Y Cf. A002865, A025065, A236912, A237667, A275972, A363226, A364345, A364465.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jul 30 2023