cp's OEIS Frontend

A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

%I A384885 #7 Jun 14 2025 23:50:59
%S A384885 1,1,1,1,2,3,4,6,8,9,13,15,18,22,28,31,38,45,53,62,74,86,105,123,146,
%T A384885 171,208,242,290,340,399,469,552,639,747,862,999,1150,1326,1514,1736,
%U A384885 1979,2256,2560,2909,3283,3721,4191,4726,5311,5973,6691,7510,8396,9395
%N A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).
%e A384885 The partition y = (8,6,3,3,3,1) has maximal anti-runs ((8,6,3),(3),(3,1)), with lengths (3,1,2), so y is counted under a(24).
%e A384885 The partition z = (8,6,5,3,3,1) has maximal anti-runs ((8,6),(5,3),(3,1)), with lengths (2,2,2), so z is not counted under a(26).
%e A384885 The a(1) = 1 through a(9) = 9 partitions:
%e A384885   (1)  (2)  (3)  (4)    (5)      (6)      (7)      (8)      (9)
%e A384885                  (3,1)  (4,1)    (4,2)    (5,2)    (5,3)    (6,3)
%e A384885                         (3,1,1)  (5,1)    (6,1)    (6,2)    (7,2)
%e A384885                                  (4,1,1)  (3,3,1)  (7,1)    (8,1)
%e A384885                                           (4,2,1)  (4,2,2)  (4,4,1)
%e A384885                                           (5,1,1)  (4,3,1)  (5,2,2)
%e A384885                                                    (5,2,1)  (5,3,1)
%e A384885                                                    (6,1,1)  (6,2,1)
%e A384885                                                             (7,1,1)
%t A384885 Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2<#1-1&]&]],{n,0,15}]
%Y A384885 For subsets instead of strict partitions we have A384177, for runs A384175.
%Y A384885 The strict case is A384880.
%Y A384885 For runs instead of anti-runs we have A384884, strict A384178.
%Y A384885 For equal instead of distinct lengths we have A384888, for runs A384887.
%Y A384885 A000041 counts integer partitions, strict A000009.
%Y A384885 A007690 counts partitions with no singletons, complement A183558.
%Y A384885 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A384885 A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
%Y A384885 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384885 A355394 counts partitions without a neighborless part, singleton case A355393.
%Y A384885 A356236 counts partitions with a neighborless part, singleton case A356235.
%Y A384885 A356606 counts strict partitions without a neighborless part, complement A356607.
%Y A384885 Cf. A008284, A047966, A242882, A287170, A325324, A325325, A329739, A356226, A356230, A356234, A384886.
%K A384885 nonn
%O A384885 0,5
%A A384885 _Gus Wiseman_, Jun 13 2025