cp's OEIS Frontend

A375401 Number of integer partitions of n whose maximal anti-runs do not all have different maxima.

%I A375401 #6 Aug 17 2024 21:50:28
%S A375401 0,0,1,1,2,3,6,7,12,16,25,33,48,63,88,116,157,204,272,349,456,581,749,
%T A375401 946,1205,1511,1904,2371,2960,3661,4538,5577,6862,8389,10257,12472,
%U A375401 15164,18348,22192,26731,32177,38593,46254,55256,65952,78500,93340,110706
%N A375401 Number of integer partitions of n whose maximal anti-runs do not all have different maxima.
%C A375401 An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.
%e A375401 The partition y = (3,2,2,1) has maximal ant-runs ((3,2),(2,1)), with maxima (3,2), so y is not counted under a(8).
%e A375401 The a(2) = 1 through a(8) = 12 partitions:
%e A375401   (11)  (111)  (22)    (221)    (33)      (331)      (44)
%e A375401                (1111)  (2111)   (222)     (2221)     (332)
%e A375401                        (11111)  (2211)    (4111)     (2222)
%e A375401                                 (3111)    (22111)    (3311)
%e A375401                                 (21111)   (31111)    (5111)
%e A375401                                 (111111)  (211111)   (22211)
%e A375401                                           (1111111)  (32111)
%e A375401                                                      (41111)
%e A375401                                                      (221111)
%e A375401                                                      (311111)
%e A375401                                                      (2111111)
%e A375401                                                      (11111111)
%t A375401 Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]
%Y A375401 For identical instead of distinct we have A239955, ranks A073492.
%Y A375401 The complement is counted by A375133, ranks A375402.
%Y A375401 The complement for minima instead of maxima is A375134, ranks A375398.
%Y A375401 These partitions have Heinz numbers A375403.
%Y A375401 For minima instead of maxima we have A375404, ranks A375399.
%Y A375401 The reverse for identical instead of distinct is A375405, ranks A375397.
%Y A375401 A000041 counts integer partitions, strict A000009.
%Y A375401 A003242 counts anti-run compositions, ranks A333489.
%Y A375401 A055887 counts sequences of partitions with total sum n.
%Y A375401 A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
%Y A375401 Cf. A034296, A115029, A141199, A279790, A358830, A358836, A374632, A374761, A375136, A375396, A375400.
%K A375401 nonn
%O A375401 0,5
%A A375401 _Gus Wiseman_, Aug 17 2024