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%I #6 May 17 2023 08:35:54
%S 0,0,0,0,1,3,3,8,9,13,18,24,24,36,41,45,57,68,72,87,95,105,131,136,
%T 149,164,199,203,232,246,276,298,335,347,409,399,467,488,567,569,636,
%U 662,757,767,878,887,1028,1030,1168,1181,1342,1388,1558,1570,1789,1791
%N Number of integer partitions of n with a unique non-mode.
%C A non-mode in a multiset is an element that appears fewer times than at least one of the others. For example, the non-modes in {a,a,b,b,b,c,d,d,d} are {a,c}.
%e The a(4) = 1 through a(9) = 13 partitions:
%e (211) (221) (411) (322) (332) (441)
%e (311) (3111) (331) (422) (522)
%e (2111) (21111) (511) (611) (711)
%e (2221) (5111) (3222)
%e (4111) (22211) (6111)
%e (22111) (41111) (22221)
%e (31111) (221111) (32211)
%e (211111) (311111) (33111)
%e (2111111) (51111)
%e (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%t nmsi[ms_]:=Select[Union[ms],Count[ms,#]<Max@@Length/@Split[ms]&];
%t Table[Length[Select[IntegerPartitions[n],Length[nmsi[#]]==1&]],{n,0,30}]
%Y For middle parts instead of non-modes we have A238478, complement A238479.
%Y For modes instead of non-modes we have A362608, complement A362607.
%Y For co-modes instead of non-modes we have A362610, complement A362609.
%Y The complement is counted by A363124.
%Y For non-co-modes instead of non-modes we have A363129, complement A363128.
%Y A000041 counts integer partitions.
%Y A008284/A058398 count partitions by length/mean.
%Y A362611 counts modes in prime factorization, triangle A362614.
%Y A363127 counts non-modes in prime factorization, triangle A363126.
%Y Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.
%K nonn
%O 0,6
%A _Gus Wiseman_, May 16 2023