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A384882 Number of integer partitions of n with all distinct lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.

%I A384882 #5 Jun 21 2025 00:41:45
%S A384882 1,1,1,2,2,3,2,5,4,5,6,9,7,12,12,11,16,18,17,25,25,23,33,35,36,42,52,
%T A384882 45,58,64,60,77,91,79,109,108,105,129,149,134,170,179,177,213,236,208,
%U A384882 275,281,282,323,359,330,410,433,440,474,541,508,614,631,635
%N A384882 Number of integer partitions of n with all distinct lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.
%e A384882 The partition (6,5,5,5,3,2) has maximal runs ((6,5),(5),(5),(3,2)), with lengths (2,1,1,2), so is not counted under a(26).
%e A384882 The partition (6,5,5,5,4,3,2) has maximal runs ((6,5),(5),(5,4,3,2)), with lengths (2,1,4), so is counted under a(30).
%e A384882 The a(1) = 1 through a(13) = 12 partitions:
%e A384882   1  2  3   4    5    6    7     8    9     A     B      C      D
%e A384882         21  211  32   321  43    332  54    433   65     543    76
%e A384882                  221       322   431  432   532   443    651    544
%e A384882                            421   521  621   541   542    732    643
%e A384882                            3211       3321  721   632    921    652
%e A384882                                             4321  821    6321   832
%e A384882                                                   4322   43221  A21
%e A384882                                                   5321          4432
%e A384882                                                   43211         5431
%e A384882                                                                 7321
%e A384882                                                                 43321
%e A384882                                                                 432211
%t A384882 Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,30}]
%Y A384882 For subsets instead of strict partitions we have A384175, equal lengths A243815.
%Y A384882 The strict case is A384178, for anti-runs A384880.
%Y A384882 Counting gaps of 0 gives A384884, equal A384887.
%Y A384882 For equal instead of distinct lengths we have A384904, strict case A384886.
%Y A384882 A000041 counts integer partitions, strict A000009.
%Y A384882 A047993 counts partitions with max part = length (A106529).
%Y A384882 A098859 counts Wilf partitions (complement A336866), compositions A242882.
%Y A384882 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384882 Cf. A008284, A047966, A089259, A325325, A382857, A383013, A383708.
%K A384882 nonn
%O A384882 0,4
%A A384882 _Gus Wiseman_, Jun 20 2025