A363129 Number of integer partitions of n with a unique non-co-mode.

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

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(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)
		

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.

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}]