A363264 - OEIS

%I #5 Jun 07 2023 08:31:56

%S 0,0,0,1,0,0,2,1,2,3,3,2,7,5,9,12,13,13,22,19,29,33,39,43,63,63,82,94,

%T 111,119,159,164,203,229,272,301,370,400,479,538,628,692,826,904,1053,

%U 1181,1353,1502,1742,1919,2205,2456,2790,3097,3539,3911,4435,4929

%N Number of integer partitions of n covering an initial interval of positive integers with a more than one 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}.

%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 A000009, complement A087897.

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

%Y The complement is counted by A363263.

%Y For mode we have A363485, complement A363484.

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

%Y A067029 counts minima in prime factorization, co-modes A362613.

%Y A071178 counts maxima in prime factorization, modes A362611.

%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, A096765, A117989, A243737, A275870, A362612.

%K nonn

%O 0,7

%A _Gus Wiseman_, Jun 06 2023