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A364464 Number of strict integer partitions of n where no part is the difference of two consecutive parts.

%I A364464 #12 Sep 26 2023 13:38:59
%S A364464 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 A364464 86,95,110,120,135,160,171,197,219,247,277,312,342,386,431,476,527,
%U A364464 598,640,727,796,893,966,1097,1178,1327,1435,1602,1740,1945,2084,2337
%N A364464 Number of strict integer partitions of n where no part is the difference of two consecutive parts.
%C A364464 In other words, the parts are disjoint from the first differences.
%e A364464 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 A364464 The a(1) = 1 through a(9) = 6 strict partitions:
%e A364464   (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)
%e A364464                  (3,1)  (3,2)  (5,1)  (4,3)  (5,3)  (5,4)
%e A364464                         (4,1)         (5,2)  (6,2)  (7,2)
%e A364464                                       (6,1)  (7,1)  (8,1)
%e A364464                                                     (4,3,2)
%e A364464                                                     (5,3,1)
%t A364464 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]=={}&]],{n,0,15}]
%o A364464 (Python)
%o A364464 from collections import Counter
%o A364464 from sympy.utilities.iterables import partitions
%o A364464 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 A364464 For length instead of differences we have A240861, non-strict A229816.
%Y A364464 For all differences of pairs of elements we have A364346, for subsets A007865.
%Y A364464 For subsets instead of strict partitions we have A364463, complement A364466.
%Y A364464 The non-strict version is A363260.
%Y A364464 The complement is counted by A364536, non-strict A364467.
%Y A364464 A000041 counts integer partitions, strict A000009.
%Y A364464 A008284 counts partitions by length, strict A008289.
%Y A364464 A120641 counts strict double-free partitions, non-strict A323092.
%Y A364464 A320347 counts strict partitions w/ distinct differences, non-strict A325325.
%Y A364464 Cf. A002865, A025065, A236912, A237667, A275972, A363226, A364345, A364465.
%K A364464 nonn
%O A364464 0,5
%A A364464 _Gus Wiseman_, Jul 30 2023