A363263 - OEIS

A363263

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

%I #5 Jun 07 2023 08:32:00

%S 0,1,1,1,2,3,2,4,4,5,7,10,8,13,13,15,19,25,24,35,35,43,50,61,59,79,83,

%T 98,111,137,137,176,187,219,240,284,298,360,385,444,485,568,600,706,

%U 763,867,951,1088,1168,1345,1453,1641,1792,2023,2179,2467,2673,2988

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

%C We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

%e The a(1) = 1 through a(10) = 7 partitions:

%e 1 11 111 211 221 21111 2221 22211 22221 33211

%e 1111 2111 111111 22111 221111 32211 222211

%e 11111 211111 2111111 2211111 322111

%e 1111111 11111111 21111111 2221111

%e 111111111 22111111

%e 211111111

%e 1111111111

%e The a(9) = 5 through a(12) = 8 partitions:

%e (22221) (33211) (33221) (2222211)

%e (32211) (222211) (222221) (3222111)

%e (2211111) (322111) (322211) (3321111)

%e (21111111) (2221111) (332111) (32211111)

%e (111111111) (22111111) (2222111) (222111111)

%e (211111111) (3221111) (2211111111)

%e (1111111111) (22211111) (21111111111)

%e (221111111) (111111111111)

%e (2111111111)

%e (11111111111)

%t comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];

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

%Y For parts instead of multiplicities we have A087897, complement A000009.

%Y For multisets instead of partitions we have A105039, complement A363224.

%Y The complement is counted by A363264.

%Y For mode we have A363484, complement A363485.

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

%Y A097979 counts normal multisets with a unique mode, complement A363262.

%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, A025147, A096765, A117989, A243737, A362612.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jun 06 2023