A363264 Number of integer partitions of n covering an initial interval of positive integers with a more than one co-mode.

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, 111, 119, 159, 164, 203, 229, 272, 301, 370, 400, 479, 538, 628, 692, 826, 904, 1053, 1181, 1353, 1502, 1742, 1919, 2205, 2456, 2790, 3097, 3539, 3911, 4435, 4929

Offset: 0

Author

Gus Wiseman, Jun 06 2023

Keywords

• nonn

Comments

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}.

Crossrefs

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

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

The complement is counted by A363263.

For mode we have A363485, complement A363484.

A000041 counts integer partitions, A000009 covering an initial interval.

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

A071178 counts maxima in prime factorization, modes A362611.

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

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

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

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

Cf. A002865, A008284, A096765, A117989, A243737, A275870, A362612.

Programs

Mathematica

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]>1&]]],{n,0,30}]