cp's OEIS Frontend

A384880 Number of strict integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

%I A384880 #6 Jun 14 2025 23:51:34
%S A384880 1,1,1,1,2,2,3,4,6,6,9,10,12,15,18,21,25,30,34,41,46,55,63,75,85,99,
%T A384880 114,133,152,178,201,236,269,308,352,404,460,525,594,674,763,865,974,
%U A384880 1098,1236,1385,1558,1745,1952,2181,2435,2712,3026,3363,3740,4151,4612
%N A384880 Number of strict integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).
%e A384880 The strict partition y = (10,7,6,4,2,1) has maximal anti-runs ((10,7),(6,4,2),(1)), with lengths (2,3,1), so y is counted under a(30).
%e A384880 The a(1) = 1 through a(14) = 18 partitions (A-E = 10-14):
%e A384880   1  2  3  4   5   6   7    8    9    A    B    C    D     E
%e A384880            31  41  42  52   53   63   64   74   75   85    86
%e A384880                    51  61   62   72   73   83   84   94    95
%e A384880                        421  71   81   82   92   93   A3    A4
%e A384880                             431  531  91   A1   A2   B2    B3
%e A384880                             521  621  532  542  B1   C1    C2
%e A384880                                       541  632  642  643   D1
%e A384880                                       631  641  651  652   653
%e A384880                                       721  731  732  742   743
%e A384880                                            821  741  751   752
%e A384880                                                 831  832   761
%e A384880                                                 921  841   842
%e A384880                                                      931   851
%e A384880                                                      A21   932
%e A384880                                                      6421  941
%e A384880                                                            A31
%e A384880                                                            B21
%e A384880                                                            7421
%t A384880 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Length/@Split[#,#2<#1-1&]&]],{n,0,30}]
%Y A384880 For subsets instead of strict partitions we have A384177.
%Y A384880 For runs instead of anti-runs we have A384178.
%Y A384880 This is the strict case of A384885.
%Y A384880 A000041 counts integer partitions, strict A000009.
%Y A384880 A047993 counts partitions with max part = length.
%Y A384880 A098859 counts Wilf partitions (complement A336866), compositions A242882.
%Y A384880 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384880 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384880 Cf. A008289, A089259, A325324, A325325, A329739, A357982, A384175, A384176, A384884, A384886.
%K A384880 nonn
%O A384880 0,5
%A A384880 _Gus Wiseman_, Jun 13 2025