cp's OEIS Frontend

A364536 Number of strict integer partitions of n where some part is a difference of two consecutive parts.

%I A364536 #9 Sep 26 2023 13:39:09
%S A364536 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,
%T A364536 106,127,146,176,205,230,277,315,366,421,483,552,640,727,829,950,1083,
%U A364536 1218,1408,1577,1794,2017,2298,2561,2919,3255,3685,4116,4638,5163
%N A364536 Number of strict integer partitions of n where some part is a difference of two consecutive parts.
%C A364536 In other words, strict partitions with parts not disjoint from first differences.
%e A364536 The a(3) = 1 through a(15) = 11 partitions (A = 10, B = 11, C = 12):
%e A364536   21  .  .  42   421  431  63   532   542   84    742   743   A5
%e A364536             321       521  621  541   632   642   841   752   843
%e A364536                                 631   821   651   A21   761   942
%e A364536                                 721   5321  921   5431  842   C21
%e A364536                                 4321        5421  6421  B21   6432
%e A364536                                             6321  7321  6431  6531
%e A364536                                                         6521  7431
%e A364536                                                         7421  7521
%e A364536                                                         8321  8421
%e A364536                                                               9321
%e A364536                                                               54321
%t A364536 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]!={}&]],{n,0,30}]
%o A364536 (Python)
%o A364536 from collections import Counter
%o A364536 from sympy.utilities.iterables import partitions
%o A364536 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
%Y A364536 For all differences of pairs we have A363226, non-strict A363225.
%Y A364536 For all non-differences of pairs we have A364346, strict A364345.
%Y A364536 The strict complement is counted by A364464, non-strict A363260.
%Y A364536 For subsets of {1..n} we have A364466, complement A364463.
%Y A364536 The non-strict case is A364467, ranks A364537.
%Y A364536 A000041 counts integer partitions, strict A000009.
%Y A364536 A008284 counts partitions by length, strict A008289.
%Y A364536 A323092 counts double-free partitions, strict A120641.
%Y A364536 A325325 counts partitions with distinct first-differences, strict A320347.
%Y A364536 Cf. A002865, A025065, A093971, A108917, A229816, A236912, A237113, A237667, A326083, A364347.
%K A364536 nonn
%O A364536 0,7
%A A364536 _Gus Wiseman_, Jul 31 2023