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A363129
Number of integer partitions of n with a unique non-co-mode.
6
0, 0, 0, 0, 1, 3, 3, 9, 12, 18, 24, 37, 43, 64, 81, 99, 129, 162, 201, 247, 303, 364, 457, 535, 653, 765, 943, 1085, 1315, 1517, 1830, 2096, 2516, 2877, 3432, 3881, 4622, 5235, 6189, 7003, 8203, 9261, 10859, 12199, 14216, 15985, 18544, 20777, 24064, 26897
OFFSET
0,6
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}.
EXAMPLE
The a(4) = 1 through a(9) = 18 partitions:
(211) (221) (411) (322) (332) (441)
(311) (3111) (331) (422) (522)
(2111) (21111) (511) (611) (711)
(2221) (3221) (3222)
(3211) (4211) (3321)
(4111) (5111) (4221)
(22111) (22211) (4311)
(31111) (32111) (5211)
(211111) (41111) (6111)
(221111) (22221)
(311111) (33111)
(2111111) (42111)
(51111)
(321111)
(411111)
(2211111)
(3111111)
(21111111)
MATHEMATICA
ncomsi[ms_]:=Select[Union[ms], Count[ms, #]>Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], Length[ncomsi[#]]==1&]], {n, 0, 30}]
CROSSREFS
For parts instead of multiplicities we have A002133.
For middles instead of non-co-modes we have A238478, complement A238479.
For modes instead of non-co-modes we have A362608, complement A362607.
For co-modes instead of non-co-modes we have A362610, complement A362609.
For non-modes instead of non-co-modes we have A363125, complement A363124.
The complement is counted by A363128.
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.
Sequence in context: A117153 A045810 A166720 * A325243 A319271 A066314
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 18 2023
STATUS
approved