cp's OEIS Frontend

A363125 Number of integer partitions of n with a unique non-mode.

%I A363125 #6 May 17 2023 08:35:54
%S A363125 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 A363125 149,164,199,203,232,246,276,298,335,347,409,399,467,488,567,569,636,
%U A363125 662,757,767,878,887,1028,1030,1168,1181,1342,1388,1558,1570,1789,1791
%N A363125 Number of integer partitions of n with a unique non-mode.
%C A363125 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 A363125 The a(4) = 1 through a(9) = 13 partitions:
%e A363125   (211)  (221)   (411)    (322)     (332)      (441)
%e A363125          (311)   (3111)   (331)     (422)      (522)
%e A363125          (2111)  (21111)  (511)     (611)      (711)
%e A363125                           (2221)    (5111)     (3222)
%e A363125                           (4111)    (22211)    (6111)
%e A363125                           (22111)   (41111)    (22221)
%e A363125                           (31111)   (221111)   (32211)
%e A363125                           (211111)  (311111)   (33111)
%e A363125                                     (2111111)  (51111)
%e A363125                                                (411111)
%e A363125                                                (2211111)
%e A363125                                                (3111111)
%e A363125                                                (21111111)
%t A363125 nmsi[ms_]:=Select[Union[ms],Count[ms,#]<Max@@Length/@Split[ms]&];
%t A363125 Table[Length[Select[IntegerPartitions[n],Length[nmsi[#]]==1&]],{n,0,30}]
%Y A363125 For middle parts instead of non-modes we have A238478, complement A238479.
%Y A363125 For modes instead of non-modes we have A362608, complement A362607.
%Y A363125 For co-modes instead of non-modes we have A362610, complement A362609.
%Y A363125 The complement is counted by A363124.
%Y A363125 For non-co-modes instead of non-modes we have A363129, complement A363128.
%Y A363125 A000041 counts integer partitions.
%Y A363125 A008284/A058398 count partitions by length/mean.
%Y A363125 A362611 counts modes in prime factorization, triangle A362614.
%Y A363125 A363127 counts non-modes in prime factorization, triangle A363126.
%Y A363125 Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.
%K A363125 nonn
%O A363125 0,6
%A A363125 _Gus Wiseman_, May 16 2023