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%I #5 May 18 2023 08:34:52
%S 0,0,0,0,1,3,3,9,12,18,24,37,43,64,81,99,129,162,201,247,303,364,457,
%T 535,653,765,943,1085,1315,1517,1830,2096,2516,2877,3432,3881,4622,
%U 5235,6189,7003,8203,9261,10859,12199,14216,15985,18544,20777,24064,26897
%N Number of integer partitions of n with a unique non-co-mode.
%C We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.
%e The a(4) = 1 through a(9) = 18 partitions:
%e (211) (221) (411) (322) (332) (441)
%e (311) (3111) (331) (422) (522)
%e (2111) (21111) (511) (611) (711)
%e (2221) (3221) (3222)
%e (3211) (4211) (3321)
%e (4111) (5111) (4221)
%e (22111) (22211) (4311)
%e (31111) (32111) (5211)
%e (211111) (41111) (6111)
%e (221111) (22221)
%e (311111) (33111)
%e (2111111) (42111)
%e (51111)
%e (321111)
%e (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%t ncomsi[ms_]:=Select[Union[ms],Count[ms,#]>Min@@Length/@Split[ms]&];
%t Table[Length[Select[IntegerPartitions[n],Length[ncomsi[#]]==1&]],{n,0,30}]
%Y For parts instead of multiplicities we have A002133.
%Y For middles instead of non-co-modes we have A238478, complement A238479.
%Y For modes instead of non-co-modes we have A362608, complement A362607.
%Y For co-modes instead of non-co-modes we have A362610, complement A362609.
%Y For non-modes instead of non-co-modes we have A363125, complement A363124.
%Y The complement is counted by A363128.
%Y A000041 counts integer partitions.
%Y A008284/A058398 count partitions by length/mean.
%Y A362611 counts modes in prime factorization, triangle A362614.
%Y A362613 counts co-modes in prime factorization, triangle A362615.
%Y A363127 counts non-modes in prime factorization, triangle A363126.
%Y A363131 counts non-co-modes in prime factorization, triangle A363130.
%Y Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.
%K nonn
%O 0,6
%A _Gus Wiseman_, May 18 2023
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