A363128 Number of integer partitions of n with more than one non-co-mode.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 6, 9, 18, 25, 44, 60, 96, 122, 188, 243, 344, 442, 615, 769, 1047, 1308, 1722, 2150, 2791, 3430, 4405, 5401, 6803, 8326, 10408, 12608, 15641, 18906, 23179, 27935, 34061, 40778, 49451, 59038, 71060, 84604, 101386, 120114, 143358

Offset: 0

Author

Gus Wiseman, May 18 2023

Keywords

• nonn

Comments

We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

Examples

			The a(9) = 1 through a(12) = 9 partitions:
  (32211)  (33211)   (33221)    (43311)
           (42211)   (52211)    (44211)
           (322111)  (322211)   (62211)
                     (332111)   (422211)
                     (422111)   (522111)
                     (3221111)  (3222111)
                                (3321111)
                                (4221111)
                                (32211111)
		

Crossrefs

For parts instead of multiplicities we have

For middles instead of non-co-modes we have A238479, complement A238478.

For modes instead of non-co-modes we have A362607, complement A362608.

For co-modes instead of non-co-modes we have A362609, complement A362610.

For non-modes instead of non-co-modes we have A363124, complement A363125.

The complement is counted by A363129.

A000041 counts integer partitions.

A008284/A058398 count partitions by length/mean.

A362611 counts modes in prime factorization, triangle A362614.

A362613 counts co-modes in prime factorization, triangle A362615.

A363127 counts non-modes in prime factorization, triangle A363126.

A363131 counts non-co-modes in prime factorization, triangle A363130.

Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.

Programs

Mathematica

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