A363124 Number of integer partitions of n with more than one non-mode.

0, 0, 0, 0, 0, 0, 0, 1, 3, 6, 9, 19, 28, 46, 65, 98, 132, 190, 251, 348, 451, 603, 768, 1014, 1273, 1648, 2052, 2604, 3233, 4062, 4984, 6203, 7582, 9333, 11349, 13890, 16763, 20388, 24528, 29613, 35502, 42660, 50880, 60883, 72376, 86158, 102120, 121133, 143010

Offset

0,9

Comments

A non-mode in a multiset is an element that appears fewer times than at least one of the others. For example, the non-modes in {a,a,b,b,b,c,d,d,d} are {a,c}.

Links

Table of n, a(n) for n=0..48.

Example

The a(7) = 1 through a(10) = 9 partitions:
  (3211)  (3221)   (3321)    (5221)
          (4211)   (4221)    (5311)
          (32111)  (4311)    (6211)
                   (5211)    (32221)
                   (42111)   (43111)
                   (321111)  (52111)
                             (322111)
                             (421111)
                             (3211111)

Mathematica

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

Crossrefs

For middle parts instead of non-modes we have A238479, complement A238478.

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

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

The complement is counted by A363125.

For non-co-modes instead of non-modes we have A363128, complement A363129.

A000041 counts integer partitions.

A008284/A058398 count partitions by length/mean.

A362611 counts modes in prime factorization, triangle A362614.

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

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

Sequence in context: A100852 A059006 A342596 * A018186 A223504 A322949

Adjacent sequences: A363121 A363122 A363123 * A363125 A363126 A363127

Keyword

nonn

Author

Gus Wiseman, May 16 2023

Status

approved