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A364467 Number of integer partitions of n where some part is the difference of two consecutive parts.

%I A364467 #10 Sep 26 2023 13:39:20
%S A364467 0,0,0,1,1,2,4,5,9,13,21,28,42,55,78,106,144,187,255,325,429,554,717,
%T A364467 906,1165,1460,1853,2308,2899,3582,4468,5489,6779,8291,10173,12363,
%U A364467 15079,18247,22124,26645,32147,38555,46285,55310,66093,78684,93674,111104
%N A364467 Number of integer partitions of n where some part is the difference of two consecutive parts.
%C A364467 In other words, the parts are not disjoint from their own first differences.
%e A364467 The a(3) = 1 through a(9) = 13 partitions:
%e A364467   (21)  (211)  (221)   (42)     (421)     (422)      (63)
%e A364467                (2111)  (321)    (2221)    (431)      (621)
%e A364467                        (2211)   (3211)    (521)      (3321)
%e A364467                        (21111)  (22111)   (3221)     (4221)
%e A364467                                 (211111)  (4211)     (4311)
%e A364467                                           (22211)    (5211)
%e A364467                                           (32111)    (22221)
%e A364467                                           (221111)   (32211)
%e A364467                                           (2111111)  (42111)
%e A364467                                                      (222111)
%e A364467                                                      (321111)
%e A364467                                                      (2211111)
%e A364467                                                      (21111111)
%t A364467 Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]!={}&]],{n,0,30}]
%o A364467 (Python)
%o A364467 from collections import Counter
%o A364467 from sympy.utilities.iterables import partitions
%o A364467 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 A364467 For all differences of pairs parts we have A363225, complement A364345.
%Y A364467 The complement is counted by A363260.
%Y A364467 For subsets of {1..n} we have A364466, complement A364463.
%Y A364467 The strict case is A364536, complement A364464.
%Y A364467 These partitions have ranks A364537.
%Y A364467 A000041 counts integer partitions, strict A000009.
%Y A364467 A008284 counts partitions by length, strict A008289.
%Y A364467 A050291 counts double-free subsets, complement A088808.
%Y A364467 A323092 counts double-free partitions, ranks A320340.
%Y A364467 A325325 counts partitions with distinct first differences.
%Y A364467 Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.
%K A364467 nonn
%O A364467 0,6
%A A364467 _Gus Wiseman_, Jul 31 2023