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A384887 Number of integer partitions of n with all equal lengths of maximal gapless runs (decreasing by 0 or 1).

%I A384887 #6 Jun 16 2025 23:50:57
%S A384887 1,1,2,3,5,6,9,10,14,18,21,26,35,39,46,58,68,79,97,111,131,155,177,
%T A384887 206,246,278,318,373,423,483,563,632,722,827,931,1058,1209,1354,1528,
%U A384887 1736,1951,2188,2475,2762,3097,3488,3886,4342,4876,5414,6038,6741,7482
%N A384887 Number of integer partitions of n with all equal lengths of maximal gapless runs (decreasing by 0 or 1).
%e A384887 The partition y = (6,5,5,5,3,3,2,1) has maximal gapless runs ((6,5,5,5),(3,3,2,1)), with lengths (4,4), so y is counted under a(30).
%e A384887 The a(1) = 1 through a(8) = 14 partitions:
%e A384887   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A384887        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A384887              (111)  (31)    (41)     (42)      (52)       (53)
%e A384887                     (211)   (221)    (51)      (61)       (62)
%e A384887                     (1111)  (2111)   (222)     (322)      (71)
%e A384887                             (11111)  (321)     (2221)     (332)
%e A384887                                      (2211)    (3211)     (2222)
%e A384887                                      (21111)   (22111)    (3221)
%e A384887                                      (111111)  (211111)   (3311)
%e A384887                                                (1111111)  (22211)
%e A384887                                                           (32111)
%e A384887                                                           (221111)
%e A384887                                                           (2111111)
%e A384887                                                           (11111111)
%t A384887 Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#,#2>=#1-1&]&]],{n,0,15}]
%Y A384887 The strict case is A384886, distinct A384178.
%Y A384887 For distinct instead of equal lengths we have A384884.
%Y A384887 For anti-runs instead of runs we have A384888, distinct A384885.
%Y A384887 For subsets instead of strict partitions we have A243815.
%Y A384887 Without counting decreases by 0 we get A384904.
%Y A384887 A000041 counts integer partitions, strict A000009.
%Y A384887 A007690 counts partitions with no singletons, complement A183558.
%Y A384887 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A384887 A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
%Y A384887 A355394 counts partitions without a neighborless part, singleton case A355393.
%Y A384887 A356236 counts partitions with a neighborless part, singleton case A356235.
%Y A384887 A356606 counts strict partitions without a neighborless part, complement A356607.
%Y A384887 Cf. A008284, A044813, A047993, A325324, A325325, A356226, A356230, A356233, A356234, A384175, A384177, A384880.
%K A384887 nonn
%O A384887 0,3
%A A384887 _Gus Wiseman_, Jun 15 2025