cp's OEIS Frontend

A384888 Number of integer partitions of n with all equal lengths of maximal anti-runs (decreasing by more than 1).

%I A384888 #6 Jun 16 2025 23:50:45
%S A384888 1,1,2,3,5,6,9,10,13,17,20,24,32,36,44,55,64,75,92,105,125,147,169,
%T A384888 195,231,263,303,351,401,458,532,600,686,784,889,1010,1152,1296,1468,
%U A384888 1662,1875,2108,2384,2669,3001,3373,3775,4222,4734,5278,5896,6576,7322
%N A384888 Number of integer partitions of n with all equal lengths of maximal anti-runs (decreasing by more than 1).
%e A384888 The partition y = (10,6,6,4,3,1) has maximal anti-runs ((10,6),(6,4),(3,1)), with lengths (2,2,2), so y is counted under a(30).
%e A384888 The a(1) = 1 through a(8) = 13 partitions:
%e A384888   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A384888        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A384888              (111)  (31)    (41)     (42)      (52)       (53)
%e A384888                     (211)   (221)    (51)      (61)       (62)
%e A384888                     (1111)  (2111)   (222)     (322)      (71)
%e A384888                             (11111)  (321)     (2221)     (332)
%e A384888                                      (2211)    (3211)     (2222)
%e A384888                                      (21111)   (22111)    (3221)
%e A384888                                      (111111)  (211111)   (22211)
%e A384888                                                (1111111)  (32111)
%e A384888                                                           (221111)
%e A384888                                                           (2111111)
%e A384888                                                           (11111111)
%t A384888 Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#,#2<#1-1&]&]],{n,0,15}]
%Y A384888 The strict case is new, distinct A384880.
%Y A384888 For distinct instead of equal lengths we have A384885.
%Y A384888 For runs instead of anti-runs we have A384887, distinct A384884.
%Y A384888 For subsets instead of strict partitions we have A384889, distinct A384177, runs A243815.
%Y A384888 A000041 counts integer partitions, strict A000009.
%Y A384888 A007690 counts partitions with no singletons, complement A183558.
%Y A384888 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A384888 A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
%Y A384888 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384888 A355394 counts partitions without a neighborless part, singleton case A355393.
%Y A384888 A356236 counts partitions with a neighborless part, singleton case A356235.
%Y A384888 A356606 counts strict partitions without a neighborless part, complement A356607.
%Y A384888 Cf. A008284, A044813, A047993, A242882, A287170, A325325, A356226, A384175, A384176, A384178, A384886.
%K A384888 nonn
%O A384888 0,3
%A A384888 _Gus Wiseman_, Jun 15 2025