login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A363124
Number of integer partitions of n with more than one non-mode.
6
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}.
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.
Sequence in context: A100852 A059006 A342596 * A018186 A223504 A322949
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 16 2023
STATUS
approved