A363125 Number of integer partitions of n with a unique non-mode.

0, 0, 0, 0, 1, 3, 3, 8, 9, 13, 18, 24, 24, 36, 41, 45, 57, 68, 72, 87, 95, 105, 131, 136, 149, 164, 199, 203, 232, 246, 276, 298, 335, 347, 409, 399, 467, 488, 567, 569, 636, 662, 757, 767, 878, 887, 1028, 1030, 1168, 1181, 1342, 1388, 1558, 1570, 1789, 1791

Offset: 0

Author

Gus Wiseman, May 16 2023

Keywords

• nonn

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}.

Examples

			The a(4) = 1 through a(9) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (5111)     (3222)
                          (4111)    (22211)    (6111)
                          (22111)   (41111)    (22221)
                          (31111)   (221111)   (32211)
                          (211111)  (311111)   (33111)
                                    (2111111)  (51111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
		

Crossrefs

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

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

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

The complement is counted by A363124.

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

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.

Programs

Mathematica

nmsi[ms_]:=Select[Union[ms],Count[ms,#]