cp's OEIS Frontend

A364062 Number of integer partitions of n with unique co-mode 1.

%I A364062 #6 Jul 13 2023 16:28:56
%S A364062 0,1,1,1,1,2,1,3,2,3,3,6,2,8,6,9,6,16,7,21,12,23,18,39,17,47,32,59,40,
%T A364062 86,44,110,72,131,95,188,103,233,166,288,201,389,244,490,347,587,440,
%U A364062 794,524,974,727,1187,903,1547,1106,1908,1459,2303,1826,2979,2198
%N A364062 Number of integer partitions of n with unique co-mode 1.
%C A364062 These are partitions with at least one 1 but with fewer 1's than each of the other parts.
%C A364062 We define a co-mode in a multiset to be an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.
%e A364062 The a(n) partitions for n = 5, 7, 11, 13, 15:
%e A364062   (221)    (331)      (551)          (661)            (771)
%e A364062   (11111)  (2221)     (33221)        (4441)           (44331)
%e A364062            (1111111)  (33311)        (33331)          (55221)
%e A364062                       (222221)       (44221)          (442221)
%e A364062                       (2222111)      (332221)         (3322221)
%e A364062                       (11111111111)  (2222221)        (3333111)
%e A364062                                      (22222111)       (22222221)
%e A364062                                      (1111111111111)  (222222111)
%e A364062                                                       (111111111111111)
%t A364062 comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
%t A364062 Table[Length[Select[IntegerPartitions[n],comodes[#]=={1}&]],{n,0,30}]
%Y A364062 For high (or unique) mode we have A241131, ranks A360013.
%Y A364062 For low mode we have A241131, ranks A360015.
%Y A364062 Allowing any unique co-mode gives A362610, ranks A359178.
%Y A364062 These partitions have ranks A364061.
%Y A364062 Adding all 1-free partitions gives A364159, ranks A364158.
%Y A364062 A000041 counts integer partitions, strict A000009.
%Y A364062 A008284 counts partitions by length (or decreasing mean), strict A008289.
%Y A364062 A237984 counts partitions containing their mean, ranks A327473.
%Y A364062 A327472 counts partitions not containing their mean, ranks A327476.
%Y A364062 A362608 counts partitions w/ unique mode, ranks A356862, complement A362605.
%Y A364062 A362611 counts modes in prime indices, triangle A362614.
%Y A364062 A362613 counts co-modes in prime indices, triangle A362615.
%Y A364062 A363486 gives least mode in prime indices, A363487 greatest.
%Y A364062 Cf. A002865, A027336, A325347, A098859, A360014, A362562, A362606, A363720, A363723, A363725, A363726.
%K A364062 nonn
%O A364062 0,6
%A A364062 _Gus Wiseman_, Jul 12 2023