cp's OEIS Frontend

A363260 Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.

%I A363260 #11 Sep 26 2023 13:39:33
%S A363260 1,1,2,2,4,5,7,10,13,17,21,28,35,46,57,70,87,110,130,165,198,238,285,
%T A363260 349,410,498,583,702,819,983,1136,1353,1570,1852,2137,2520,2898,3390,
%U A363260 3891,4540,5191,6028,6889,7951,9082,10450,11884,13650,15508,17728,20113
%N A363260 Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.
%e A363260 The a(1) = 1 through a(8) = 13 partitions:
%e A363260   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A363260        (11)  (111)  (22)    (32)     (33)      (43)       (44)
%e A363260                     (31)    (41)     (51)      (52)       (53)
%e A363260                     (1111)  (311)    (222)     (61)       (62)
%e A363260                             (11111)  (411)     (322)      (71)
%e A363260                                      (3111)    (331)      (332)
%e A363260                                      (111111)  (511)      (611)
%e A363260                                                (4111)     (2222)
%e A363260                                                (31111)    (3311)
%e A363260                                                (1111111)  (5111)
%e A363260                                                           (41111)
%e A363260                                                           (311111)
%e A363260                                                           (11111111)
%t A363260 Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]=={}&]],{n,0,30}]
%o A363260 (Python)
%o A363260 from collections import Counter
%o A363260 from sympy.utilities.iterables import partitions
%o A363260 def A363260(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), 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 A363260 For length instead of differences we have A229816, strict A240861.
%Y A363260 For all differences of pairs parts we have A364345.
%Y A363260 For subsets of {1..n} instead of partitions we have A364463.
%Y A363260 The strict case is A364464.
%Y A363260 A000041 counts integer partitions, strict A000009.
%Y A363260 A008284 counts partitions by length, strict A008289.
%Y A363260 A323092 counts double-free partitions, ranks A320340.
%Y A363260 A325325 counts partitions with distinct first-differences.
%Y A363260 Cf. A002865, A025065, A108917, A236912, A237113, A237667, A320347, A326083, A363225, A364347.
%K A363260 nonn
%O A363260 0,3
%A A363260 _Gus Wiseman_, Jul 19 2023