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Number of integer partitions of n whose minima of maximal anti-runs are not all different.
10

%I #6 Aug 17 2024 21:50:33

%S 0,0,1,1,3,3,7,9,14,19,30,38,56,73,102,133,179,231,307,392,511,647,

%T 831,1046,1328,1658,2084,2586,3219,3970,4909,6016,7386,9005,10988,

%U 13330,16175,19531,23580,28350,34067,40788,48809,58215,69383,82461,97917,115976

%N Number of integer partitions of n whose minima of maximal anti-runs are not all different.

%C An anti-run is a sequence with no adjacent equal terms. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.

%C Also the number of reversed integer partitions of n such that the minima of maximal anti-runs are not all different.

%e The a(0) = 0 through a(8) = 14 reversed partitions:

%e . . (11) (111) (22) (113) (33) (115) (44)

%e (112) (1112) (114) (223) (116)

%e (1111) (11111) (222) (1114) (224)

%e (1113) (1123) (1115)

%e (1122) (1222) (1124)

%e (11112) (11113) (1133)

%e (111111) (11122) (2222)

%e (111112) (11114)

%e (1111111) (11123)

%e (11222)

%e (111113)

%e (111122)

%e (1111112)

%e (11111111)

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

%Y The complement for maxima instead of minima is A375133, ranks A375402.

%Y The complement is counted by A375134, ranks A375398.

%Y These partitions are ranked by A375399.

%Y For maxima instead of minima we have A375401, ranks A375403.

%Y For identical instead of distinct we have 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, A046660, A073492, A115029, A141199, A239955, A279790, A358830, A358836, A375136.

%K nonn

%O 0,5

%A _Gus Wiseman_, Aug 17 2024