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Number of integer partitions of n whose maximal anti-runs do not all have different maxima.
10

%I #6 Aug 17 2024 21:50:28

%S 0,0,1,1,2,3,6,7,12,16,25,33,48,63,88,116,157,204,272,349,456,581,749,

%T 946,1205,1511,1904,2371,2960,3661,4538,5577,6862,8389,10257,12472,

%U 15164,18348,22192,26731,32177,38593,46254,55256,65952,78500,93340,110706

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

%C 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 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 The a(2) = 1 through a(8) = 12 partitions:

%e (11) (111) (22) (221) (33) (331) (44)

%e (1111) (2111) (222) (2221) (332)

%e (11111) (2211) (4111) (2222)

%e (3111) (22111) (3311)

%e (21111) (31111) (5111)

%e (111111) (211111) (22211)

%e (1111111) (32111)

%e (41111)

%e (221111)

%e (311111)

%e (2111111)

%e (11111111)

%t Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]

%Y For identical instead of distinct we have A239955, ranks A073492.

%Y The complement is counted by A375133, ranks A375402.

%Y The complement for minima instead of maxima is A375134, ranks A375398.

%Y These partitions have Heinz numbers A375403.

%Y For minima instead of maxima we have A375404, ranks A375399.

%Y The reverse for identical instead of distinct is A375405, ranks A375397.

%Y A000041 counts integer partitions, strict A000009.

%Y A003242 counts anti-run compositions, ranks A333489.

%Y A055887 counts sequences of partitions with total sum n.

%Y A375128 lists minima of maximal anti-runs of prime indices, sums A374706.

%Y Cf. A034296, A115029, A141199, A279790, A358830, A358836, A374632, A374761, A375136, A375396, A375400.

%K nonn

%O 0,5

%A _Gus Wiseman_, Aug 17 2024