A363484 - OEIS

%I #8 Jun 07 2023 08:31:52

%S 0,1,1,1,2,3,2,5,6,6,8,11,12,17,20,21,27,35,38,50,56,65,76,95,105,125,

%T 146,167,198,233,252,305,351,394,457,522,585,681,778,878,994,1135,

%U 1269,1446,1638,1828,2067,2339,2613,2940,3301,3684,4143,4634,5156,5771

%N Number of integer partitions of n covering an initial interval of positive integers with a unique mode.

%C A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

%e The a(1) = 1 through a(8) = 6 partitions:

%e   (1)  (11)  (111)  (211)   (221)    (21111)   (2221)     (3221)

%e                     (1111)  (2111)   (111111)  (3211)     (22211)

%e                             (11111)            (22111)    (32111)

%e                                                (211111)   (221111)

%e                                                (1111111)  (2111111)

%e                                                           (11111111)

%t Table[If[n==0,0,Length[Select[IntegerPartitions[n], Union[#]==Range[Max@@#]&&Length[Commonest[#]]==1&]]],{n,0,30}]

%Y For parts instead of multiplicities we have A096765, complement A025147.

%Y For multisets instead of partitions we have A097979, complement A363262.

%Y For co-mode we have A363263, complement A363264.

%Y The complement is counted by A363485.

%Y A000041 counts integer partitions, A000009 covering an initial interval.

%Y A362607 counts partitions with multiple modes, co-modes A362609.

%Y A362608 counts partitions with a unique mode, co-mode A362610.

%Y A362614 counts partitions by number of modes, co-modes A362615.

%Y Cf. A002865, A008284, A105039, A117989, A243737, A362612.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jun 05 2023